#index divisor

[algebraic-dumbass]

YOU. ME. FIBER PRODUCT (Z' x Z') What are we representing today? pi1(Z') of course! Uh oh, looks like the kernel of the action of pi1(Z') on the fibers of the fiber product has finite index, we form the finite covering space T^{n+1}. we do it again with the epimorphism pi1(T^{n+1}) -> H^1(T{n+1},Z/2Z). You know what that means. PULLBACK (W) TIME! now consider the diagonal section of the fiber product. What does it do? It maps Z' to a divisor S(Z'). DIVISOR, ACTED (AS(Z')). DIVISOR, ADDED (D = S(Z') + AS(Z')). DIVISOR, pulled back to W? oh yes, please. We pull back the divisor to our pullback W. then we prove its cohomology class is even, even as in divisible by 2. DIVISIBLE? DIVISOR GIVING A COVERING RAMIFIED ALONG IT? UH, I THINK SO next thing you know, this concludes the induction step. Then i pull back the classes, evaluate them on these bundles, black out, wake up, do a bump, white out which i didn't know you could do, then i smonked a joint, greened-out, then the classes were nonzero, uh oh, looks like the math is kicking in-

ijijvpokqscokqcnebk bu guj opkcomd jvklr kb o opjidjcj b b

[souhaillaghchimdev]

Python: 100 Simple Codes

Python: 100 Simple Codes

Beginner-friendly collection of easy-to-understand Python examples.

Each code snippet is designed to help you learn programming concepts step by step, from basic syntax to simple projects. Perfect for students, self-learners, and anyone who wants to practice Python in a fun and practical way.

Codes:

1. Print Hello World

2. Add Two Numbers

3. Check Even or Odd

4. Find Maximum of Two Numbers

5. Simple Calculator

6. Swap Two Variables

7. Check Positive, Negative or Zero

8. Factorial Using Loop

9. Fibonacci Sequence

10. Check Prime Number

===

11. Sum of Numbers in a List

12. Find the Largest Number in a List

13. Count Characters in a String

14. Reverse a String

15. Check Palindrome

16. Generate Random Number

17. Simple While Loop

18. Print Multiplication Table

19. Convert Celsius to Fahrenheit

20. Check Leap Year

===

21. Find GCD (Greatest Common Divisor)

22. Find LCM (Least Common Multiple)

23. Check Armstrong Number

24. Calculate Power (Exponent)

25. Find ASCII Value

26. Convert Decimal to Binary

27. Convert Binary to Decimal

28. Find Square Root

29. Simple Function

30. Function with Parameters

===

31. Function with Default Parameter

32. Return Multiple Values from Function

33. List Comprehension

34. Filter Even Numbers from List

35. Simple Dictionary

36. Loop Through Dictionary

37. Check if Key Exists in Dictionary

38. Use Set to Remove Duplicates

39. Sort a List

40. Sort List in Descending Order

===

41. Create a Tuple

42. Loop Through a Tuple

43. Unpack a Tuple

44. Find Length of a List

45. Append to List

46. Remove from List

47. Pop Last Item from List

48. Use range() in Loop

49. Use break in Loop

50. Use continue in Loop

===

51. Check if List is Empty

52. Join List into String

53. Split String into List

54. Use enumerate() in Loop

55. Nested Loop

56. Simple Class Example

57. Class Inheritance

58. Read Input from User

59. Try-Except for Error Handling

60. Raise Custom Error

===

61. Lambda Function

62. Map Function

63. Filter Function

64. Reduce Function

65. Zip Two Lists

66. List to Dictionary

67. Reverse a List

68. Sort List of Tuples by Second Value

69. Flatten Nested List

70. Count Occurrences in List

===

71. Check All Elements with all()

72. Check Any Element with any()

73. Find Index in List

74. Convert List to Set

75. Find Intersection of Sets

76. Find Union of Sets

77. Find Difference of Sets

78. Check Subset

79. Check Superset

80. Loop with Else Clause

===

81. Use pass Statement

82. Use del to Delete Item

83. Check Type of Variable

84. Format String with f-string

85. Simple List Slicing

86. Nested If Statement

87. Global Variable

88. Check if String Contains Substring

89. Count Characters in Dictionary

90. Create 2D List

===

91. Check if List Contains Item

92. Reverse a Number

93. Sum of Digits

94. Check Perfect Number

95. Simple Countdown

96. Print Pattern with Stars

97. Check if String is Digit

98. Check if All Letters Are Uppercase

99. Simple Timer with Sleep

100. Basic File Write and Read

===

[jamesmilleer2407]

S&P 500 Index Movements and Their Connection to Economic Shifts

The S&P 500 Index has long served as a foundational benchmark for the U.S. equity market. As one of the most recognized indices in global finance, it represents a broad cross-section of American companies across multiple sectors. Created by Standard & Poor’s in 1957, the index tracks the performance of 500 large-cap firms and reflects trends that influence economic and corporate activity in the United States. Its composition and behavior offer insight into the shifting dynamics of the business environment and investor sentiment.

What the S&P 500 Index Represents

The S&P 500 Index is widely viewed as a barometer of the U.S. stock market. Covering companies from sectors including technology, energy, healthcare, consumer goods, industrials, and financials, the index captures approximately 80% of the total market capitalization of U.S. equities. It includes well-established corporations that are often regarded as leaders in their industries, and its diversified nature makes it useful for observing general trends across the economy. The inclusion of firms in varying stages of business maturity ensures that it reflects both cyclical movements and long-term shifts.

Criteria for Inclusion in the Index

In order to be listed in the S&P 500 Index, companies must meet a defined set of eligibility requirements. These include a minimum level of market capitalization, adequate liquidity, and a specific proportion of publicly available shares. Additionally, firms must be headquartered in the U.S. and should have reported positive earnings in the most recent quarters. These standards are enforced by a committee that routinely evaluates candidates and existing members to ensure that the index remains representative of large-cap U.S. equities.

This selective process means the index reflects financially stable, publicly traded companies with consistent performance. The rigorous inclusion criteria also contribute to its widespread use as a standard of measurement for market-wide performance.

Weighting and Calculation Method

The S&P 500 Index is weighted by market capitalization, meaning larger companies have a greater influence on the S&P 500 Index movements. The weighting is adjusted by float, which accounts for the shares that are available for public trading. The index’s value is determined by summing the float-adjusted market capitalizations of its components and dividing the result by a proprietary divisor maintained by S&P Dow Jones Indices.

This method ensures that shifts in share prices of the most valuable companies drive overall changes in the index more significantly than smaller firms. As a result, a surge or drop in share value of one of the largest firms can influence the broader index, even when the majority of smaller companies remain stable.

Sector Distribution and Economic Insight

Sector distribution within the S&P 500 Index changes over time depending on performance and valuation shifts. Technology and healthcare often make up a substantial portion of the index, reflecting the economic emphasis on innovation and medical advancement. At other times, energy, financials, or industrials may become more heavily weighted based on sector growth or contraction.

This dynamic composition allows the index to offer a view into sector leadership within the U.S. economy. Market analysts often observe which sectors are gaining momentum within the index to understand broader economic direction and sentiment shifts. The prominence of a sector within the S&P 500 Index can indicate where business activity is concentrated or expanding.

Comparison to Other Market Indices

The S&P 500 Index is frequently compared with other indices such as the Dow Jones Industrial Average and the Nasdaq Composite. While the Dow includes just 30 companies and is price-weighted, the S&P 500 Index provides broader exposure and a more balanced perspective through its market-cap weighted structure. The Nasdaq Composite, which is known for its emphasis on technology companies, includes thousands of stocks, many of which are smaller or more volatile than those in the S&P 500 Index.

Due to its wider representation and reliable methodology, the S&P 500 Index is often referred to in economic discussions and market analysis. Its historical performance has often aligned closely with general trends in corporate revenue growth, inflation, and consumer activity.

Usage in Financial Products and Market Observation

Many financial products are based on or linked to the S&P 500 Index. Index funds and exchange-traded funds (ETFs), for instance, track the performance of the index to provide exposure to its components. The structure of these products allows market participants to observe broad trends without the need to monitor individual companies.

Beyond financial products, the index is a primary reference point for media outlets, analysts, and corporate leaders. It serves as a metric to compare corporate performance, evaluate market conditions, and monitor macroeconomic shifts. Movements in the index are often discussed in relation to policy changes, earnings seasons, or global economic events.

Limitations of the Index

While the S&P 500 Index offers a robust view of the U.S. market, it is not without its limitations. One notable limitation is its emphasis on large-cap stocks. Mid- and small-cap companies are excluded, which means that certain emerging businesses and regional trends may be underrepresented. Additionally, the weighting method can sometimes result in an outsized influence of just a few companies, particularly during periods of market concentration.

Sector imbalances can also affect interpretation. During certain periods, one or two sectors may dominate the S&P 500 Index, which could lead to skewed performance readings that are not reflective of the broader economy. These factors suggest the importance of looking beyond a single index when analyzing long-term trends.

The S&P 500 Index continues to play a central role in tracking market direction, measuring corporate performance, and understanding sector dynamics. Its construction, weighting, and broad industry coverage make it a valuable resource for understanding how various parts of the economy are performing. Though not all-encompassing, it remains one of the most widely recognized and referenced tools for analyzing the health and momentum of the U.S. stock market.

[postpalette]

Fintechzoom FTSE 100: A Comprehensive Analysis of Market Trends and Performance

The Fintechzoom 100 is one of the most prominent stock indices globally, representing the 100 largest companies listed on the London Stock Exchange (LSE). Investors, traders, and financial analysts closely follow its movements to gauge the economic health of the UK market. Platforms like Fintechzoom provide real-time insights, news, and analysis on FTSE 100 trends, helping investors make informed decisions.

In this article, we will delve into the Fintechzoom FTSE 100 market, its performance trends, key drivers, and how investors can benefit from its fluctuations.

Understanding the FTSE 100 Index

The FTSE 100 (Financial Times Stock Exchange 100) index consists of the 100 largest publicly traded companies by market capitalization in the UK. It serves as a benchmark for measuring the overall performance of the London Stock Exchange. Some well-known companies in the FTSE 100 include HSBC, BP, Unilever, Shell, and AstraZeneca.

How is the Fintechzoom 100 Calculated?

The Fintechzoom 100 is a market capitalization-weighted index, meaning companies with a higher market cap have a greater impact on the index’s overall performance. The calculation involves:

Market capitalization: The total value of a company's outstanding shares.

Index divisor: Adjusted periodically to maintain consistency in the index value.

Stock prices: A weighted average of all listed companies' share prices.

Why Fintechzoom is Crucial for FTSE 100 Analysis

Fintechzoom is a leading financial news and analysis platform that provides real-time updates, expert insights, and detailed reports on the Fintechzoom 100 index. Investors and traders rely on Fintechzoom for the latest market trends, economic indicators, and corporate earnings reports.

Key Features of Fintechzoom for FTSE 100 Tracking

Live Market Data:

Real-time FTSE 100 index movements, stock performance, and sector-wise analysis.

Expert Analysis:

In-depth market insights and trend forecasts from financial experts.

Economic News:

Coverage of macroeconomic factors impacting FTSE 100 stocks.

Investment Tips:

Guidance on portfolio diversification, risk management, and trading strategies.

Factors Affecting the Fintechzoom 100 Performance

Several key factors influence the movements of the FTSE 100 index, including:

1. Economic Data and GDP Growth

The overall performance of the UK economy, measured by GDP growth, directly impacts the Fintechzoom 100. A growing economy boosts investor confidence, leading to a rise in stock prices.

2. Interest Rates and Inflation

The Bank of England's monetary policy, including interest rate changes, affects investor sentiment. Higher interest rates generally lead to lower stock prices, whereas lower rates stimulate economic growth and drive up stock prices.

3. Global Events and Geopolitical Tensions

Events such as Brexit, trade wars, or geopolitical conflicts significantly influence investor confidence. Any uncertainty in global markets can lead to volatility in the Fintechzoom 100.

4. Corporate Earnings and Financial Reports

Quarterly earnings reports of FTSE 100-listed companies play a crucial role in shaping investor sentiment. Positive earnings often lead to stock price increases, while poor earnings can result in declines.

5. Currency Exchange Rates (GBP/USD and GBP/EUR)

The strength of the British Pound (GBP) relative to other currencies can impact multinational companies listed in the Fintechzoom 100. A weaker GBP can benefit exporters, while a stronger GBP may hurt their profits.

Recent FTSE 100 Performance Overview

1. FTSE 100’s Growth in 2024

In recent months, the Fintechzoom 100 has witnessed significant fluctuations due to economic and geopolitical uncertainties. The market started strong in early 2024, with sectors like energy, finance, and technology driving gains.

2. Impact of Inflation on FTSE 100 Stocks

High inflation levels have led to concerns over corporate profitability. However, sectors like oil and gas have benefited from rising commodity prices. Fintechzoom provides detailed analysis on inflation’s effect on individual stocks.

3. Sector-Wise Performance

Energy Sector: Companies like Shell and BP have outperformed due to higher oil prices.

Financial Sector: Banks such as HSBC and Barclays have seen mixed performance due to interest rate hikes.

Technology Sector: Growth in tech stocks has been influenced by global trends and AI advancements.

How Investors Can Benefit from FTSE 100 Movements

1. Long-Term Investment Strategy

Investing in Fintechzoom 100 companies for the long term can yield stable returns. Blue-chip stocks like Unilever, AstraZeneca, and Diageo offer dividends and steady growth potential.

2. Short-Term Trading Opportunities

Day traders and swing traders monitor Fintechzoom FTSE 100 updates to capitalize on market volatility. Strategies such as momentum trading and technical analysis help traders make informed decisions.

3. ETF Investments

Exchange-Traded Funds (ETFs) tracking the Fintechzoom 100 allow investors to gain exposure to the entire index with lower risk. ETFs like iShares FTSE 100 UCITS ETF provide diversified investment options.

4. Dividend Investing

Several Fintechzoom 100 companies offer attractive dividend yields. Investing in dividend stocks ensures passive income, making it a popular strategy among long-term investors.

Future Predictions for the FTSE 100

According to financial experts at Fintechzoom,  is expected to remain volatile in 2024 due to ongoing macroeconomic uncertainties. However, potential growth areas include:

Technology and AI-driven stocks

Renewable energy investments

Financial sector recovery post-inflation control

Conclusion  Fintechzoom 100

The Fintechzoom 100 remains a crucial indicator of the UK’s economic performance. With platforms like Fintechzoom, investors can stay updated on market trends, economic news, and investment opportunities. Whether you are a long-term investor or a short-term trader, understanding the Fintechzoom 100 movements and key factors affecting it can help in making informed financial decisions.

#Fintechzoom100

[qocsuing]

Understanding the US30 Index: A Barometer of the U.S. Economy

Understanding the US30 Index: A Barometer of the U.S. Economy The US30, also known as the Dow Jones Industrial Average (DJIA), is one of the most widely watched stock market indices in the world . It tracks the performance of 30 large, publicly traded companies in the United States, providing a snapshot of the overall health of the U.S. economy.To get more news about what is us30, you can visit our official website.

Key Features of the US30 Index Composition: The US30 includes 30 prominent companies from various sectors, such as technology, healthcare, and finance . These companies are selected by a committee and are considered to be the most influential in the U.S. economy . Calculation: The index is price-weighted, meaning that companies with higher stock prices have a greater impact on the index's value . The value is calculated by adding up the stock prices of the 30 companies and dividing by a divisor, which is adjusted for stock splits and other corporate actions . Historical Significance: Created by Charles Dow in 1896, the US30 is the second-oldest stock market index in the U.S . It has been a reliable indicator of the stock market's performance for over a century

Economic Indicator: The US30 is often used as a barometer of the U.S. economy . When the index is up, it generally indicates positive economic conditions, while a decline suggests economic challenges.

How to Invest in the US30 Investors can gain exposure to the US30 through various financial instruments, such as exchange-traded funds (ETFs) and mutual funds . Some popular options include the SPDR Dow Jones Industrial Average ETF (DIA) and the iShares Dow Jones U.S. ETF (IYY) . Criticisms and Alternatives Despite its long history and widespread use, the US30 has faced criticism for not being fully representative of the broader U.S. market . Unlike market-cap-weighted indices like the S&P 500, the US30 includes only 30 large-cap companies, which may not reflect the performance of smaller companies . In conclusion, the US30 index remains a valuable tool for investors and economists to gauge the health of the U.S. economy. While it has its limitations, its long-standing history and influence make it an important benchmark in the financial world.

[ezyforextrading]

What is Sensex? Sensex, a portmanteau of "Sensitive" and "Index," was coined by Deepak Mohoni, a stock market expert. It represents the most popular market index of 30 companies listed under the Bombay Stock Exchange (BSE). These component companies are among the largest and most actively traded stocks in India. Criteria for Inclusion The selection of companies under the Sensex is governed by the S&P BSE Index Committee, which adheres to specific criteria: Listing on the Bombay Stock Exchange in India. Large or mega-cap stocks. Liquidity of the stocks. Earnings generated from core activities. Contribution to maintaining sector balance in the country’s equity market. Evolution of Sensex Since its inception in the early 1990s, Sensex has witnessed remarkable growth, particularly post-2000. Various milestones mark its journey, reflecting the dynamic nature of India's economic landscape. Milestones Early 90’s to the end of the 20th century July 1990: Sensex closed at 1001, touching 1000 for the first time. 1991-1992: Introduction of liberal economic policies led to crossing the 2000 mark. 1999: Crossed 5000 points, marking the onset of the new century. Beginning of the 21st century to mid-2000s IT boom propelled the index beyond 6000 points. Settlement in the Ambani family spurred growth, crossing 7000 points. Rapid expansion from 9000 to 10,000 points between June and December 2005. Mid-2000s to its end Touched 10,003 points in February 2006. Rapid growth to 20,000 points by December 2007. Closed at 21,004.96 points in November 2010, surpassing the 21,000 mark. 2013-2019 Closed at 21,033.97 points in October 2013. Crossed 30,000 points in January 2015. Surged past 40,000 points in May 2019. Methodology of Calculation Free-Float Market Capitalization The calculation of the Sensex relies on the free-float market capitalization of its 30 constituent companies. This methodology considers only shares available for trading, excluding those held by promoters, governments, and strategic investors. Weighted Methodology Sensex follows a market capitalization-weighted methodology, wherein the weightage of each stock in the index is determined by its free-float market capitalization. Factors Affecting Calculation Several factors influence the calculation of the Sensex, including: Market capitalization changes of constituent companies. Stock price fluctuations. Adjustments due to bonus issues, stock splits, mergers, and acquisitions. Sensex Formula The formula for calculating Sensex is: Sensex = (Total free-float market capitalization of 30 companies / Index Divisor) x Base Value Understanding Sensex vs. Nifty Sensex and Nifty are both crucial benchmark indices of the Indian stock market, each with distinct features and calculations. Sensex Represents the top 30 companies on the BSE. Calculated using a market capitalization-weighted methodology. Nifty Comprises the top 50 companies on the National Stock Exchange (NSE). Calculated using a free-float market capitalization-weighted methodology. Investing in Sensex Investing in Sensex involves several steps: Opening a Demat Account: to hold shares electronically. Opening a Trading Account: for buying and selling securities online. Holding a Bank Account: necessary for trading activities. Major Plunges in Sensex Sensex has weathered significant market downturns, notably during the global financial crisis of 2008-2009. Instances include: Loss of 1408 points on 21st January 2008, the highest since inception. Consistent drops throughout 2008, hitting a low of 8509.56 points in October. Nearly 750 points decline in 2009 due to the Satyam fraud. FAQs What is Sensex in simple words? Sensex, short for Sensitive Index, represents a basket of stocks from 30 large and actively traded companies listed on the Bombay Stock Exchange (BSE), providing a snapshot of India's stock market performance. How Sensex works? Sensex is calculated

using the free-float market capitalization method, which considers the market valueof only the freely tradable stocks of the index constituents. Why is Sensex used? Sensex serves as a barometer of the Indian stock market, reflecting investor sentiment and overall economic health. Why does the Sensex fluctuate? Fluctuations in Sensex are influenced by various factors including economic indicators, geopolitical events, corporate earnings, and investor sentiment. How do I purchase Sensex shares? Investors can purchase shares represented in the Sensex by opening a Demat and trading account, followed by buying shares through the stock exchange. https://ezyforextrading.com/learn-trading/sensex/?feed_id=4003&_unique_id=65c51229e251a&EzyForexTrading

[versilperihelion]

c) well here we are again, it's always such a pleasure

yeah there's potentially a lot of terms. but, we've done this several times, so we know how it goes. we keep the term that has the highest power of 10 on it, leave the rest of the terms aside for later, and then we add and subtract terms that have the further higher powers of 10 so all the digits are once again in the same order as the powers. so let's do exactly that once again, and keep in mind the c_rest for later.

so let's split c_n and rearrange it.

cool, let's start with the bigger part.

here we can extract the common factor 10^2k and we get our original number.

one half of the sequence done. what about the rest?

i named it c_smaller because with the minuses it'd be a smaller number, not because it's a smaller thing to write out. each of the two sequences has matching terms with the same x on it so we can pair them up.

yeah it's still unwieldy. let's look at each term there separately, starting from the first three.

looks like a pattern. what about the last three terms?

yep, it's a pattern. for every term with index i, looks like it would be

we can't rely on the digits x_i to give us the divisibility by 37 cause they're, well, single-digit, and 37 is a prime anyway. we can't count on the power of 10 cause it only has various arrangements of 2 and 5 as its divisors, and again, 37 is prime. leaving us once again with some power of 10 minus 1. and it looks like we should have everything we need from all we did before, but before we generalize it, let's make a step to the side and prove something that you will likely be like "well duh" when i write it down.

------

interlude: if you take 10 to the power k where k is any whole number bigger than 0, and you subtract 1 from it, the resulting number is a string of k nines. woof.

"well duh", right? it's obvious when you think about it, and it makes sense, so why are we wasting time on this? well, the question is how do you approach proving it formally, and can we get something useful out of it? let's try and see. and for this we'll use the trick called proof by induction, which means we put the proof on a stove and it heats nicely but doesn't burn our hands if we touch it.

joke aside what proof by induction means is that if you have something you want to prove for all natural numbers starting from 0 or 1 or whatever, you just need to do two things: first, prove it for the first number. then, prove that if it works for any one number, then it must work for the very next one. you could also expand this to something other than just natural numbers, as long as there's some ordering so it makes sense to say that one element is the very next after the previous.

so let's prove it for k=1, which is gonna be as obvious as it sounds.

a bit silly but it works. we subtracted one, bumped the power down by one, we had the power k=1 and we have 1 nine in the end. math hasn't failed us.

now let's see what happens for any number k as the power.

aha, after we bumped it down a power and split it into 9+1 like we did 1000=999+1 a while back, and out popped how you'd write the output for the previous number in the sequence. so, each next item is the previous one plus a nine on the next power of ten up from the last, all the way down to r_1 that makes the very first nine.

we get the power k-1 after expanding r_k, then power k-2 after expanding r_k-1, and so on. and because we have k terms (from 1 to k-1 and 0), we have k nines in a row when we write it out in base-10. so we proved the obvious thing formally.

so why did we do this again? well we can now without any hesitation generalize what we did before to our c_smol items:

so, we know that 999 works for this, and 999,999 does too. it definitely feels like each next number here should work as well. and it's true - because we already proved it today. if you take any number divisible by 37, such as 999, and insert somewhere in the middle a sequence of three of the same digit, such as another 999, you get another number divisible by 37. and because for each i-th item this number has 3i digits, all of them are just 999 to which each time we insert another 999, so every one of those numbers is divisible by 37.

so where were we again? right, we had that new number c_n, where we showed that c_bigger is a multiple of 37 cause we got our starting number out of it, and then from that we subtract c_smaller, which is a sum of all those c_smol_i items. and now we proved that each of those items is divisible by 37. so we are summing and subtracting a lot of items, all of which are divisible by 37, so the final result must also be divisible by 37. and with that, we're done.

------

so, what have we accomplished? we proved that any number divisible by 37 has those three fancy properties for finding a new number divisible by 37, by exploiting the fact that it's actually 111 doing all those things, and 111 itself is divisible by 37.

but here's the secret: this is not really a property of 111.

as in, it's not the property of the number one hundred and onety-one. i mean one hundred and eleven.

it is, in fact, the property of any number that can be written as "111" in any numerical base.

all those times we had 10 to some power? we could put there any base instead of 10 and express the number in that base. wherever we had 999 pop up, we would have three of the highest digit in that base - in base-6 it'd be 555, in base-8 it'd be 777, in base-12 it'd be BBB (where B=11), in base-16 it'd be FFF (where F=15).

and there when we did that interlude of proving that 10 to some power minus 1 gives a string of as many nines as the power? what we did there was substitute 10-1 for 9, being the highest single-digit. in base-16, where "10" would mean sixteen, one less than that is F (fifteen), the highest single-digit number. i told you it'd be useful.

this means we can expand the three properties: a) if you take a three-digit number in any base that is a multiple of "111" or of any divisor of "111", if you shuffle the digits one to the left, you get another multiple of that same number b) if you take any number in any base that is a multiple of "111" or any of its divisors, if you insert a sequence of three of any digit (that exists in this base) somewhere in the middle, you get another multiple of that number c) if you take any number in any base that is a multiple of "111" or any divisor of "111", if you reverse the digits and intersperse them with zeroes, you get another multiple of that same number

let's go through some example, and let's go with base-16.

so we can expect that this will work if we use 3, 7, 13, or any multiple of those as our equivalent to 37.

all those numbers would work. let's pick 39, just cause, and prove that very first property, remembering that wherever we had powers of 10, we now need powers of 16.

so we make a new number, same as before.

we've done this song and dance several times already, we add and subtract x_2 times third power of our base and collect terms based on common factors.

the first bit is our starting number times 16, as expected. the second bit looks like how we got 999 originally. how about now?

and that's that. i don't think there's any greater moral in this. i guess that understanding the notation you use can lead you to some discoveries? breaking a problem into smaller pieces can help tackle the bigger problem? there's always some neat math to be found? dunno, find one, i just wrote all that cause i promised two people i would, and in the process found that yeah this works in any base, just with different numbers. i tried to sneak a digital root joke somewhere in there given how many times 999 showed up but couldn't find anything so i'll just write 3421 and that's the end of this post.

alright i don't tend to post stuff if it's not already part of an existing thread cause i don't believe people would care but i promised @demifiendcruithne i would write this up as a separate thing if anyone asked about it, and well @pteren did so here we are.

if you come across this and don't like math, this is your cue to move on to the next thing on your dash.

the context is that for pteren's month-long physical challenge i decided to also find some fun properties for each daily number of reps i do. and for 37 there were three such properties:

if you take any 3-digit number that is divisible by 37, and you shuffle its digits one space to the left (so e.g. 148 → 481), you'll get another number divisible by 37 (yes shuffling them to the right also works but it's the same as going left twice so it's just one of those cases that matters)

if you take any number divisible by 37 and insert into it a string of three of the same digit (so e.g. 148 → 147778, or 166648), you'll get another number divisible by 37

if you take any number divisible by 37, reverse it, and add a zero between every digit (so e.g. 148 → 80401 or 1147 → 7040101), you guessed it, you'll get another number divisible by 37

so i said there's already enough math i'm putting into the physical challenge thread so if anyone's interested in how to prove something like that, i'd put it into a separate post. so here we are. this is what i got myself into.

note: there will be quite a lot of images here. all of them, every single one, will have some math formulae and nothing else, cause it'll be much easier to read if that math will be neatly formatted and such using latex (the math kind, not the kinky kind) rather than somewhat put into plaintext of a text post. all those images will have added alt text but i have no idea how to best do image descriptions for that, so i'm kinda winging it. i know i said i doubt people would read this, but i'm still not gonna half-ass trying to maintain accessibility.

in fact there are enough images that i need to split it into the main post and a rb with the second half cause the limit is apparently 30. if for some unholy reason you came across this before i paste in and format the second half, and for some even more unholy reason you decide it's something you want to rb, please just wait until i add the second half. if there was a way to make a draft and an immediate rb of said draft, i would've done it.

also note: i'll be using the letter i as the index in a sequence. there will be no imaginary unit mentioned anywhere or needed for anything. there's no calculus either, the most complex arithmetic involved is knowing that if you multiply two powers of the same base, you get the sum of the exponents (2^5 x 2^2 = 32 x 4 = 128 = 2^7). and even better than that, basically every single number featured and labeled with a letter will be a positive whole number. we're not even going into fractions here.

also also note: i'm not gonna try to be 100% rigorous here. that's for academia, i was just bored with hellsite access.

so, given this includes some fuckery on the digits of a number, i think best to start by something we don't consciously think about too often: what does it mean for a number to look some way in base 10? let's take, say, number 1438:

1438 = 1 x 10^3 + 4 x 10^2 + 3 x 10^1 + 8 x 10^0

almost exactly the same way we'd pronounce this in english, it's one thousand (10 to the third power), four hundreds (10 to the second power), three tens (10 to the first power), and eight ones (10 to the zeroth power - anything other than 0 raised to 0th power gives 1. we are not getting into why not 0 here).

so in general we can write what this means, symbolically:

where x_0 is the ones digit, x_1 is the tens, and so on, which can also fancily be written using the sigma notation which if you have any programming experience, is basically just how in math you'd write a for loop: for i that goes from 0 to k (including k), where k is any natural number you feel like, add together the digit at the i-th position times 10 to the i-th power.

kinda seems like overcomplicating it, but this can be applied to any number base other than 10 (and the capital sigma notation helps avoid writing too much because laziness is the source of inventions).

many people will already have known all this, but it's still useful foundation to lay down before we start fiddling with the digits, so we are actually aware of what we're doing. now we can move on to the first thing to prove.

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a) the first thing was that if we take a 3-digit number that is divisible by 37, such as 148 (it's 4 x 37), and we shuffle the digits one space to the left, wrapping back to the ones space, what we get is another number divisible by 37, such as 481 (it's 13 x 37. leet).

so let's write down some general number we're starting from.

i called this number a_s because it's the number we start with for proof a). subscript s for start, n for new.

x_2, x_1, and x_0 are just the digits in the hundreds, tens, and ones places. this starting number is a multiple of 37, so it's 37 times some natural number n. we don't really need to know or care what n is, only that it's a whole number. it's all whole number turtles all the way down.

how would we then write a_n, the new number we create from our shenanigans?

should make sense. we move the tens digit to hundreds, ones to tens, and hundreds to ones. cool. but how do we go about proving it's divisible by 37? we can't really reason much about what we have there so far, so we should try to rearrange it so we can find several parts in there that each is divisible by 37 - a multiple of some number plus another multiple of it gives you yet another multiple of that number, like 30+70=100, 30 is 6x5, 70 is 14x5, and 100 is (6+14) x 5, so 20x5. and same if we were to subtract them. so lets pull a fast one on that number and add and subtract the same thing to it.

(at this point i must also note that the interface for making a new post is incredibly stupid for short wide images where the cancel button is overlapping the button where you go to add image description. hellsite, pay your qa department more)

we added and subtracted 1000 times x_2 so we still have the same number. what was that for? if you look at the first three terms, they are in descending order in both which digit it is, and what power of 10 they're multiplied by, which looks just like our original number, but all times 10. meanwhile, the other two terms both have x_2 on them. so we can extract the common factors there.

in the parentheses of the first part we have that original number a_s, so this whole first half is divisible by 37, cause that was our premise. the other half is some single-digit number x_2, multiplied by 1000-1. so by 999. well, it just so happens that 999 is a multiple of 37, being 9x111, and 111 is 3x37. so we have

so here's the secret: this is not really a property of 37.

it's a property of 111. the reason it works for 37 is exactly because 111 is a multiple of 37, so when it works for 111, 37 just tags along for the ride. in fact you might note that it works the same for the other divisor of 111, 3. as you may know, a number is divisible by 3 if (when written in base-10) the sum of its digits is divisible by 3. when we shuffle the digits around, the sum doesn't change, so the new number is still divisible by 3.

and you can show the same way that the other two properties work for 3 just as well as they do for 37. if we reverse the digits and sprinkle some zeroes in, the sum stays the same, so new number is still a multiple of 3. if we insert three of the same digit somewhere in the middle, we're adding 3 times something into the sum, so the new sum is still divisible by 3. sneaky.

but let's show that the other two properties work for 37. spoiler alert: 111 is the key player in those as well.

------

b) if you take a multiple of 37, and insert a string of three of the same digit somewhere in the middle, the new number will also be a multiple of 37.

new year, new me, new starting number, just that it oddly looks exactly the same. except this time we are not restricted to just 3 digits of the starting number, so we need to consider any number of digits, so the laziness notation comes in

so how would we write the new number?

so i don't know if there is a proper notation but i went for square brackets here to mean "i'm not multiplying those together, those are just digits arranged from highest power to lowest". so we have the original digits going up from index 0 (right to left cause that's how we write numbers), and then three of this new digit y plopped in the middle, displacing the l-th digit and each next one up to index k three spaces to the left.

next we can write this as a sum as before, but because there's more terms it'll be useful to immediately split it into smaller, more manageable pieces.

because we added the three extra digits, everything to the left was effectively multiplied by a thousand, so the powers of ten there each grew by 3, while everything to the right is unchanged. so, as before, if we can show that each of these three pieces independently is divisible by 37, we'll have succeeded. and we'll start with the inserted part, b_insert, cause it's the simplest and i imagine some of yall already see where it's going.

that one was simple, it was straight out 111 times that digit y times some power of ten. so that's one place where 111 showed up. now, we can probably somehow get the original number somewhere out of the remaining two pieces to leave us with just one last bit to prove, if the previous proof is any indication. b_rest looks like it'd be useful but it's missing some terms, so let's look at b_front. there we'd have a neat match if not for all the powers of 10 being 3 bigger than the corresponding index. so, let's extract the common factor from all those.

now the stuff in parentheses is what we'd need to add to b_rest to complete the original number. it's just that there's a thousand of it, when we need just one. so let's separate one from the thousand and-

oh hey, and what we're left with is 999 times something, which we already know is a multiple of 37. let's call it 37 times m where m is stuff in brackets times the pieces of 999 after we pulled 37 out of it. cause we don't need to write it any more than that if we already know it's good. "i don't care" is a very powerful statement to use in math.

and now we can put those back together again.

and as a final cherry on top,

yay. so, just one left. should be smooth sailing, right?

yeah to be honest for this next one i needed some time to get one specific insight, and the road there lead through first proving something simpler, which is quite an effective problem solving approach. let's go through it and see.

------

c) if you take a number that's a multiple of 37, reverse the order of digits, and then put a zero between every two digits, you get another multiple of 37.

so a good place to start will be once again defining what we mean in math symbols, cause while we can put it into words and write an example, those math symbols have been quite useful in solving this.

nothing new here but this time i also wrote the sequence of digits cause it'll be useful in building up the new number. so we flip the order and then add zeros.

so reading this from the right, in the ones place we have x_k. then we skip tens. then in the hundreds we have x_k-1. we skip thousands, and in ten thousands we have x_k-2. that's 10^0, 10^2, 10^4, and so on that have a digit from the original sequence - the even powers of ten. the number of original digits didn't change, but we doubled each power of ten used, so the highest power should be 2k.

yeah it got a bit more complicated, with the indices all the other way, and the exponents doubled, nice mess we got ourselves into. if you can already see where the proof is going, that's great. quoting Grant Sanderson of youtube channel 3blue1brown, pause and ponder. if you don't, fret not, it took me a while too. so how about for now we ignore this bigger problem and solve a much simpler case.

------

d2) same as point c, but we're looking at two-digit starting numbers. there is no point d1, i gave it a 2 cause 2 digits.

there are only two numbers that are 2-digit and are multiples of 37 - 37 itself and 74, so we could just plop 703 and 407 into a calculator and see what they divide into, but we should do it in a general way like we did before. we're already deep into the math, might as well

so, we swap the digits, double the exponents, and add a 0 in the middle just for now to remember it was there.

last time we were in a situation like this (back in case a) we added and subtracted one of the digits times 1000. let's do it again.

looks familiar. let's extract 10^2 from the first two terms and -x_1 from the other two.

yep, that worked, and we got the original number and 111 in there. that's promising. let's try three digits.

------

d3) same as d2 but the starting number has 3 digits

okay, so last time, and back in case a, we were able to restore the original number in the correct ordering of digits by adding and subtracting one term, and we did that in d2 as well. here we'd need to add and subtract two terms each. you know, worth a shot.

so we can do that, and the stuff we subtract each can pair with one of the remaining terms. let's do the usual and see what we're left with

so we got two of those as freebies, what about the last one? 10^6 is one million, so the brackets equal 999,999. is this a multiple of 999? why yes, it is, and the comma used to bunch digits of large numbers into groups of three helps us notice. and, well, so does saying the number out loud. 999 thousand, 999.

alright, i think we're warmed up enough to get back to our general case

which as i mentioned before i need to split into a separate followup post. 2/10 literally unplayable. i know there is the option "post privately" and there is the queue and schedule, but i'm sure as hell not gonna experiment when if something goes wrong i need to either redo the whole thing or have to wait to add the second half or whatever, i just want to drop it and move on.

[fin-markets]

Dow Jones Industrial Average (DJIA)

Also known as Dow 30, in a stock market index that tracks 30 large, publicly traded, blue chip companies trading on the New York Stock Exchange and Nasdaq. The index was created in 1896 by Charles Dow, along with his business partner Edward Jones, and thus, the index is named after them. It is the second oldest index in the US, the oldest being Dow Jones Transportation Average (DJTA)- created in 1884.

A blue chip is a nationally recognized, well-established, and financially sound company. Blue-chip companies are known to weather downturns and operate profitably in the face of adverse economic conditions, which helps to contribute to their long record of stable and reliable growth.

In the early 20th century, the overall growth of the economy was tied to the industrial sector, this cemented the relationship between Dow's performance and the growth of the US economy. This belief continues till today with man investors considering a strong performing Dow to be an indication of a strong economy.

DJIA is a price-weighted index, the companies with large market capitalization are given a place in the Index, and those who become less relevant to the Index are dropped and replaced. Market Capitalization is a method to estimate the value of a company by multiplying the number of total outstanding shares by the stock pice.

The value of the DJIA is calculated with the help of the Dow Divisor, which is a predetermined constant that is used to determine the effect of a one-point move in any of the approximately 30 stocks that comprise the Dow. The Dow Divisor as of April 20, 2022 was 0.15172752595384, which is below 0 - technically making it a multiplier. The value of DJIA is ascertained using the following formula-

DJIA Price = SUM / Dow Divisor

Here, SUM refers to sum of the price of one share of stock for all the components (30 companies).

Initially the Index had only 12 companies , but by 1928 it had included 30 companies (most of them being industrial). Since then, the composition of the DJIA has changed its composition many times as the companies that don't meet the listing criteria are replaced by those that do.

~Lakshya Kapoor

[wisdomrays]

TAFAKKUR: Part 241

MUSLIM CONTRIBUTIONS TO MATHEMATICS: Part 1

When we talk about Muslim contributions to mathematics we are usually referring to the years between 622 and 1600 ce. This was the golden era of Islam when it was influential both as a culture and religion, and was widespread from Anatolia to North Africa, from Spain to India.

Mathematics, or "the queen of the sciences" as Carl Friedrich Gauss called it, plays an important role in our lives. A world without mathematics is unimaginable. Throughout history, many scholars have made important contributions to this science, among them a great number of Muslims. It is beyond the scope of a short article like this one to mention all the contributions of Muslim scholars to mathematics; therefore, I will concentrate on only four aspects: translations of earlier works, and contributions to algebra, geometry, and trigonometry. In order to understand fully how great were the works of scholars in the past, one needs to look at them with the eye of a person of the same era, since things that are well-known facts today might not have been known at all in the past.

There has never been a conflict between science and Islam. Muslims understand everything in the universe as a letter from God Almighty inviting us to study it to have knowledge of Him. In fact, the first verse of the Qur'an to be revealed was:

Read! In the Name of your Lord, Who created… (Alaq 96:1).

Besides commanding us to read the Qur'an, by mentioning the creation the verse also draws our attention to the universe. There are many verses which ask Muslims to think, to know, to learn and so on. Moreover, there are various sayings of the Prophet Muhammad, peace be upon him, encouraging Muslims to seek knowledge. One hadith says, "A believer never stops seeking knowledge until they enter Paradise" (al-Tirmidhi).

In another hadith, the Prophet said, "Seeking knowledge is a duty on every Muslim" (Bukhari). Hence it is no surprise to see early Muslim scholars who were dealing with different sciences.

TRANSLATIONS

Prophet Muhammed (pbuh) said, “Knowledge is the lost property of a Muslim; whoever finds it must take it” ; hence Muslims started seeking knowledge. One way they did this was to start translating all kinds of knowledge that they thought to be useful. There were two main sources from which Muslim scholars made translations in order to develop the field of science, the Hindus and the Greeks. The Abbasid caliph al-Mamun (804–832) had a university built and ordered its scholars to translate into Arabic many works of Greek scholarship. Between 771 and 773 CE the Hindu numerals were introduced into the Muslim world as a result of the translation of Sithanta from Sanskrit into Arabic by Abu Abdullah Muhammad Ibrahim al-Fazari. Another great mathematician, Thabit ibn Qurra, not only translated works written by Euclid, Archimedes, Apollonius, Ptolemy and Eutocius, but he also founded a school of translation and supervised many other translations of books from Greek into Arabic. While Hajjaj bin Yusuf translated Euclid’s Elements into Arabic, al-Jayyani wrote an important commentary on it which appears in the Fihrist (Index), a work compiled by the bookseller Ibn an-Nadim in 988. A simplified version of Ptolemy’s Almagest appears in Abul-Wafa’s book of Tahir al-Majisty and Kitab al-Kamil. Abu’l Wafa Al-Buzjani commented on and simplified the works of Euclid, Ptolemy and Diophantus. The sons of Musa bin Shakir also organized translations of Greek works.

These translations played an important role in the development of mathematics in the Muslim world. Moreover, the ancient Greek texts have survived thanks to these translations.

ALGEBRA AND GEOMETRY

The word "algebra" comes from "Al-Jabr", which is taken from the title of the book Hisab Al-Jabr wal Muqabala by Muhammad ibn Musa al-Khwarizmi (780–850). Al-Khwarizmi, after whom the "algorithm" is named, was one of the great mathematicians of all times. Europe was first introduced to algebra as a result of the translation of Khwarizmi's book into Latin by Robert Chester in 1143. The book has three parts. The first part deals with six different types of equations:

(ax2 = bx) ; (ax2 = b) ; (ax = b) ; (ax2 + bx = c) ; (ax2 + c = bx) ; (bx + c = ax2)

Khwarizmi gives both arithmetic and geometric methods to solve these six types of problems. He also introduces algebraic multiplication and division. The second part of Hisab Al-Jabr deals with mensuration. Here he describes the rules of computing areas and volumes. Since Prophet Muhammad, peace be upon him, said, “Learn the laws of inheritance and teach them to people, for that is half of knowledge," the last and the largest part of this section concerns legacies, which requires a good understanding of the Islamic laws of inheritance. Khwarizmi develops Hindu numerals and introduces the concept of zero, or “sifr” in Arabic, to Europe. The word “zero” actually comes from Latin “zephirum,” which is derived from the Arabic word “sifr.”

The three sons of Musa bin Shakir (about 800–860) were perhaps the first Muslim mathematicians to study Greek works. They wrote a great book on geometry, Kitab Marifat Masakhat Al-Ashkal (The Book of the Measurement of Plane and Spherical Figures), which was later translated into Latin by Gerard of Cremona. In the book, although they used similar methods to those of Archimedes, they move a step further than the Greeks to consider volumes and areas as numbers, and hence they developed a new approach to mathematics. For example, they described the constant number pi as “the magnitude which, when multiplied by the diameter of a circle, yields the circumference.”

A well-known poet, philosopher and astronomer Omar Khayyam (1048–1122) was at the same time a great mathematician. His most famous book on algebra is Treatise on the Demonstration of Problems of Algebra. In his book besides giving both arithmetic and geometric solutions to second degree equations he also describes geometric solutions to third degree equations by the method of intersecting conic sections. He also discovered binomial expansion [26]. His work later helped develop both algebra and geometry.

Thabit bin Qurra (836–901) was an important mathematician who made many discoveries in his time. As mentioned in the Dictionary of Scientific Biography he “played an important role in preparing the way for such important mathematical discoveries as the extension of the concept of number to (positive) real numbers, integral calculus, theorems in spherical trigonometry, analytic geometry, and non-Euclidean geometry. In astronomy Thabit was one of the first reformers of the Ptolemaic system, and in mechanics he was a founder of statics.”

To give an idea of his importance, we will just give here, without details, one of his theorems on amicable numbers. Two natural numbers m and n are called “amicable” if each is equal to the sum of the proper divisors of the other:

for n > 1, let pn=3.22n–1 and qn=9.22n–1–1. If pn–1 , pn and qn are prime numbers, then a=2n pn–1 pn and b=2nqn are amicable.

[indexcalculations-blog]

How Index Calculation is Maintained

The statistical device which summarizes a group of data in a unique post number. Robotics Indexes (indices) additionally do and down movement of commercial production, and the market costs of bonds, commodities, shares, etc. Index Calculation Agent supports the entity that calculates and publishes the amount of the Index, which is considered normally Standard & Poor’s Financial Services LLC. Any corporate business, whether it requires divisor changes or not, will be implemented after the close of trading on the day earlier to the ex-date of such corporate actions, or once the Index Calculation Agent normally handles such company actions. A person who calculates the value of a derivative and/or determines who owes what to whom in a swap. The calculation agent is usually a market producer. Sometimes each party uses one calculation agent for dealing, but sometimes each party uses a different agent. In such a case, these 2 calculation agents share responsibilities and should work together on the dealings. Artificial Intelligence Indices

A Custom Index is an analytical test of the change in a securities market based on a fixed number of deals particularly preferred and weighted according to the author. The index represents the collective performance of the author's stock selections, removals, and weightings across time. The Custom Index allows buyers to quickly leverage our existing infrastructure and website applications for further processes. Indexes can use numerous factors to ensure that constituents properly influence the index according to their weight. The custom index services give you the chance to finely co-operate and work with one provider for all your indexing needs, thereby lowering cost and complexity.

One of the most DBA responsibilities is to make sure databases to perform optimally. The most efficient way to do this is through indexes. We review their information resources jobs, making sure their server is in a safe place before we dig into their work points. Indxx develops custom indices for our customers upon demand. For the index maintenance process, it is important to fix the fragmentation. This all is a little tricky process which is done very nicely at our website that is indxx.com. We help you to increase your share and profit please go and check our website for more information related to the shares.

Source: https://indexcalculations.blogspot.com/2019/10/how-index-calculation-is-maintained.html

[eigenshit]

i’m working on my abstract algebra takehome midterm, and it’s uh,, pretty fucking hard!!! but i’ve gotten through most of one question, and i gotta say, my new favorite theorems are:

the 4th isomorphism theorem

the theorem that if the index of a subgroup H in a finite group G is equal to the smallest prime divisor of G, then H is normal in G

the third sylow theorem

an absurdly useful couple of results about characteristic subgroups (which dummit&foote state without proof right after the definition of characteristic, but they’re easy to prove; see d&f 3e, 4.4, exercises 7, 8ab, if you’re interested)

[modelsbytalias]

Know About the Different Types of Excel Function

Introduction:

Using the criteria you specify, the FILTER function "filters" a set of data. The output is a list of numbers that are exact matches within the given range. In layman's terms, the excel filter function applies one or more logical checks to a data set to get matching records. Since arguments and various forms of criterion-based formulae are typically included in logical tests, they are provided. Data from a specified year or month, including a certain string, or exceeding a certain threshold can all be matched using FILTER.

Excel's "Count" Function:

Number counts are what you get back from count function in excel. Negative numbers, percentages, dates, times, fractions, and formulas yielding numbers are all examples of numbers. Cells with no data or text values are disregarded. The COUNT function counts the number of times a given input is a number. Value1, value2, value3, etc. are all valid inputs for the COUNT function. Up to 255 arguments may be used, all of which may be single hardcoded values, cell references, or ranges.

Excel's Indexing Capability:

The excel index function finds the value in a range or array at a specific index. Individual values or entire rows and columns can be retrieved with INDEX. When combined with INDEX, the MATCH function can supply row and column numbers.

Excel filter function:

The excel filter function sorts data according to user-specified criteria and returns only the relevant entries. Using the criteria you specify, the FILTER function "filters" a set of data. The output is a list of numbers that are exact matches within the given range. In layman's terms, the FILTER function applies one or more logical checks to a data set to get matching records.

Excel Sumproduct:

The excel sumproduct function calculates the total product of a set of coordinates or an array of coordinates. Although multiplication is selected by default, other operations such as addition, subtraction, and division are also feasible.

Excel roundup:

Excel roundup function will return a rounded value to the specified precision. ROUNDUP rounds all numbers up, while conventional rounding only rounds down values less than 5.

Excel offset:

If you give excel offset function five parameters—a beginning cell, an offset in rows, an offset in columns, a height in rows, and a width in columns—it will return a reference to a range with those parameters as its components. When a calculation calls for a range of values, OFFSET comes in handy.

Excel concatenate:

Excel concatenate function can connect or concatenate up to 30 values into a single string for output. The CONCAT and TEXTJOIN functions in Excel 2019 and later are superior and more versatile options.

Excel mod function:

Excel mod function calculates the difference between two numbers.  The sign of the divisor is preserved in the MOD result. The MOD operation requires two inputs, a numeric value, and a denominator. Division requires two numbers: the divisor and the dividend.

Choose function excel:

Returns a single item from the list of value arguments using the given index_num. Depending on the index number, choose function excel can be used to select one of up to 254 possible options.

[skytrust7]

Difference Between Sensex and Nifty 50

Difference Between Sensex and Nifty 50

Investing in the stock market is a great way to grow your wealth over time. But before diving into the market, it's important to understand some key concepts, such as stock market indices. Two of the most popular indices in India are the Sensex and the Nifty 50. In this blog, we'll explore the differences between these two indices.

What is the Sensex?

The Sensex, short for Sensitive Index, is the oldest stock market index in India. It was first introduced in 1986 and is composed of the 30 largest and most actively traded companies on the Bombay Stock Exchange (BSE). The Sensex is often used as a barometer of the overall health of the Indian stock market and the Indian economy. It is calculated by taking the total market capitalization of the 30 constituent companies and dividing it by a number called the "index divisor."

What is the Nifty 50?

The Nifty 50, on the other hand, is a stock market index introduced in 1996 by the National Stock Exchange of India (NSE). It consists of the 50 largest and most actively traded companies on the NSE. Like the Sensex, the Nifty 50 is considered a benchmark index for the Indian stock market and the Indian economy. It is calculated by taking the total market capitalization of the 50 constituent companies and dividing it by a number called the "index divisor."

Differences between the Sensex and Nifty 50

Composition of Index

One of the most significant differences between the Sensex and Nifty 50 is their composition. The Sensex is composed of 30 companies listed on the BSE, while the Nifty 50 consists of 50 companies listed on the NSE. While some companies are listed on both exchanges, there are still some differences in the composition of the two indices. For example, the Sensex has more companies from the manufacturing sector, while the Nifty 50 has more companies from the services sector.

Methodology of Calculation

The Sensex and Nifty 50 are calculated using different methodologies. The Sensex is calculated based on the free-float market capitalization of its constituent companies, which means that only the shares available for public trading are taken into account. On the other hand, the Nifty 50 is calculated using the full market capitalization of its constituent companies, which means that all outstanding shares, including those held by promoters and governments, are taken into account.

Historical Performance

Another difference between the Sensex and Nifty 50 is their historical performance. Since the Sensex was introduced before the Nifty 50, it has a longer history and has seen more market cycles. However, the Nifty 50 has outperformed the Sensex in recent years, thanks in part to its more diverse composition and calculation methodology.

Impact of Companies

Lastly, the impact of individual companies on the two indices varies. Since the Sensex has only 30 companies, the performance of each constituent company has a higher impact on the index as a whole. On the other hand, since the Nifty 50 has 50 companies, the impact of each constituent company on the index is relatively lower.

Conclusion

In conclusion, the Sensex and Nifty 50 are two of the most widely followed stock market indices in India. While they share some similarities, such as their role as benchmarks for the Indian stock market and economy, they differ in their composition, calculation methodology, historical performance, and impact of individual companies. As an investor, understanding these differences can help you make better-informed investment decisions and choose the index that best suits your investment goals and risk profile.

[seemoonslite]

What is a Stock Market Index?

A index of the market is a measure of performance of a market. Some indexes are only tracking the biggest companies in the country, while others cover an array of stocks. The Dow Jones Industrial Average, for instance, is comprised of 30 companies and is a measure of the average price of their shares over a specific time period ผลดาวโจนส์วันนี้. When stocks are added or subtracted from the index the divisor is adjusted to reflect this.

There are two basic types of indexes priced-weighted and market-cap-weighted. MMI is a market-cap weighted, gives more weight to large companies. The Dow Jones Industrials is the most common price-weighted index, and the Nasdaq Composite is a market-cap-weighted index.

Different weightings can be assigned to indexes, based on how they were constructed and the way each stock has changed. Although an unmanaged index isn't an investment directly, it can be a valuable source of information for investors. Certain indexes have more weight for large companies while others give more weight for smaller companies with lower prices.

A index of the market is an important tool for investors and analysts alike. It provides investors and analysts with a snapshot of the market's sentiment and details on individual stocks' performance. Indexes let investors compare stocks according to their volatility and sector. Indexes can help you to know the current state of the market and help you identify a possible buying opportunity.

The Dow Jones Industrial Average (DJIA) is a price-weighted indicator that represents approximately a quarter of all stock market value in the United States. The DJIA does not indicate an increase or decrease in market value, but rather a small variation in a huge amount of stocks.

The three most well-known stock market indexes in the USA are the S&P 500 and Dow Jones Industrial Average ดาวน์โจนส์ล่าสุดวันนี้. These indices are utilized by financial professionals and investors to assess the return on specific investments. They can also serve as a reference point for portfolios. Some investment funds even follow the stock market index when managing their portfolios.

Indexes are a popular tool for investors, and are a crucial part of the investment management business. Many investors utilize these indices to compare their performance against other fund managers. The S&P 500, as an example, represents the largest U.S. stocks by market capitalization. The Nasdaq 100 which tracks technology stocks, is a great option for investors who prefer to invest in indexes that are passive.

[congduc-1612]

Cách sử dụng hàm MOD lấy phần dư trong Excel nhanh chóng mà bạn cần biết

Hàm MOD trong Excel là gì? Đây là khá đơn giản và thông dụng, thường sử dụng để lấy phần dư cho phép toán chia trong bảng tính Excel rất thuận lợi cho bạn khi làm việc. Quá trình làm việc với bảng tính Excel, có rất nhiều bài toán yêu cầu phải lấy phần số dư của phép toán nằm thực hiện những yêu cầu tính toán khác. Vậy bạn đã biết hàm MOD có thể giúp bạn làm được điều này? Hàm này có cấu trúc ra sao? Hãy cùng NTP tìm hiểu qua bài viết được tình dưới đây nhé!

Hàm MOD trong Excel

Hàm MOD dùng để lấy phần dư của phép toán chia trong bảng tính Excel. Kết quả khi hàm Excel này trả về luôn cùng dấu với số chia.

Công thức hàm Excel

=MOD(Number, Divisor)

Giải thích hàm Excel:

Number: Số bị chia trong quá trình thực hiện phép toán. (Bắt buộc)

Divisor: Số chia trong quá trình thực hiện phép toán. (Bắt buộc)

Lưu ý khi dùng hàm Excel:

Nếu số bị chia bằng 0, hàm MOD trong Excel sẽ trả về kết quả giá trị lỗi #DIV/0!.

Trong Excel hàm MOD và INT thường được dùng để giải quyết các yêu cầu này, bạn có thể sử dụng bằng các số hạng của hàm INT để giải quyết yêu cầu bài toán

=MOD(n,d) = n-d * INT(n/d)

Bước 1: Bôi đen bảng cần thực hiện tô màu trong Excel.

Bước 2: Tại Tab Home – Chọn thẻ Conditional Formatting – Chọn mục New Rule…

Bước 3: Trong hộp thoại New Formatting Rule – Chọn mục Use a formula to determine which cells to format.

Bước 4: Tại vị trí Edit the Rule Description – Sử dụng công thức hàm MOD để thiết lập, nhập công thức =MOD(ROW(A1),2)=0 vào ô như hình dưới đây.

Bước 5: Để chọn kiểu chữ, màu chữ, màu nền đánh dấu đã được yêu cầu bạn nhấp chuột vào ô Format và lựa chọn màu thỏa yêu cầu bài toán.

Hàm index match là gì

Hàm index match trong excel dùng để dò tím giá trị trên cả cột và dòng trong bảng tính, với hàm excel khác như vlookup hay hlookup chỉ dò tìm được trên cột hoặc hàng.

Hàm index kết hợp match là hàm excel khá thông dụng và thường được dùng trong một số bài toán dò tìm giá trị phức tạp, khi nhận được yêu cầu bài toán bạn nên cân nhắc đê lựa chọn phương pháp giải cho phù hợp để nâng cao hiệu suất công việc.

Công thức hàm index excel

=INDEX(Array, Row_number, [Column_number])

Giải thích hàm:

Array: Mảng chứa đựng giá trị trả về.

Row_number ( bắt buộc): Vị trí của hàng chứa giá trị được trả về.

[Column_number]: Vị trí của cột chứa giá trị được trả về.

Công thức hàm match excel

=MATCH(Lookup_value, Lookup_array, [Match_type])

Giải thích hàm:

Lookup_value: Giá trị thực hiện việc tìm kiếm.

Lookup_array: Mảng chứa giá trị được tìm kiếm.

[Match_type]: Lựa chọn kiểu tìm kiếm mong muốn. Với [Match_type] = 1  hoặc bỏ: Hàm sẽ hiểu tìm giá trị lớn nhất mà giá trị đó nhỏ hơn Lookup_value; Với [Match_type] = 0: Hàm sẽ hiểu tìm giá trị thứ nhất mà giá trị đó bằng chính xác Lookup_value; Với [Match_type] = -1: Hàm sẽ hiểu tìm giá trị nhỏ nhất mà giá trị đó bắt buộc phải lớn hơn hoặc bằng với Lookup_value.

Hàm weekday excel

Hàm weekday excel được biết là hàm trả về thứ trong tuần tương ứng với nó, luôn là một số nguyên dương.

Chức năng của hàm weekday trong excel

Giúp người dùng có thể tối ưu được công việc khi người dùng nhập liệu ngày tháng nhưng chỉ hiển thị số lượng ngày. Ngoài ra, nó còn giúp học sinh giả guyết các yêu cầu trong bài tập được giáo viên giao làm.

Công thức weekday

=WEEKDAY(Serial_number, [Return_type])

Giải thích hàm excel:

Serial_number (bắt buộc): Giá trị ngày tháng cần chuyển về thứ.

Return_type (tùy chọn): Tham số lựa chọn theo yêu cầu.

Các dạng Return_type của hàm weekday excel

Return_type = 1: Tương ứng với 1 là chủ nhật và 7 là thứ 7.

Return_type = 2: Tương ứng với 1 là thứ 2 và 7 là chủ nhật.

Return_type = 3: Tương ứng với 0 là thứ 2 và 6 là chủ nhật.

Return_type = 11: Tương ứng với 1 là thứ 2 và 7 là chủ nhật.

Return_type = 12: Tương ứng với 1 là thứ 3 và 7 là thứ 2.

Return_type = 13: Tương ứng với 1 là thứ 4 và 7 là thứ 3.

Return_type = 14: Tương ứng với 1 là thứ 5 và 7 là thứ 4.

Return_type = 15: Tương ứng với 1 là thứ 6 và 7 là thứ 5.

Return_type = 16: Tương ứng với 1 là thứ 7 và 7 là thứ 6.

Return_type = 17: Tương ứng với 1 là chủ nhật và 7 là thứ 7.

[triviallytrue]

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Zero is even because it can be expressed in the form 0 = 2 * n where n is an integer. 2 * 0 = 0.

Zero is not prime, because primes are divisible only by themselves and 1. Divisibility means 0 / x = y for integers x and y, and for any positive value of x, 0 / x = 0, so all positive integers divide 0. That's a lot more than 2 divisors.

"Being a natural number" is not a well-defined mathematical property the way evenness or primality are. It shouldn't be used in proofs because it's unclear. I've had professors define the natural numbers as the positive integers or as the non-negative integers, and both are "valid" - commonly used and accepted.

I prefer to consider 0 a natural number because I think the non-negative integers is more of a mouthful than the positive integers, so it's better to have a shorthand for it, and because 0 is a wonderfully useful number to have in your back pocket when possible.

I've also become a CS person since studying math and everything in CS (rightfully) indexes from 0, not from 1, so that's another point in favor of 0's membership in the naturals to me.