#zero sig figs

[sig-fig-counter]

There are no significant figures in this post.

So any math nerd or anyone who studies/teaches math has probably had this type of interaction:

Random person: so what do you do?

Mathematician: I study/work with/teach math.

Random person: Oh I HATE math.

And I, my fellow nerds, have found the perfect response: You hate math? You think you even know what math is? Have you ever had to prove that numbers are real? Or that shapes exist?? Have you ever had to sit in a topology class listening to a professor and watching the students literally lose their minds??? You haven't? Then stfu, I donzt wanna hear you say a word about math.

[allthecastlesonclouds]

such a band such a band. and here you can see all my musings about albums, logos, and my shitty handwriting :D listen. it's a band ever!! i love them!!

@settlingforskeletons- you said you might want to see this, and here it do be!

[er-cryptid]

Patreon | Ko-fi    

[kusaka6e]

TUTOR

two | three | four

chapter list

———

you'd been at the library for about an hour, doing homework for your other classes while you waited for 4:30 to hit.

someone dramatically sitting in the chair across from you makes you jump, pulling one of your earbuds out.

baji is sitting in front of you, slamming a notebook down onto the table.

"did you have to make such an entrance?"

"didn’t wanna be late.”

"i didn't know you cared about being on time."

he scoffs at your comment, pulling out a pencil.

"what are you doing in chem right now?"

"i have no fucking clue." he replies blankly

you choke back a laugh at the deadpan look on his face, knowing he's being completely serious.

"you have any notes to look at?" he shakes his head, making you roll your eyes.

"no wonder you're failing, you don't have anything to study with."

"my teacher talks too fast for me to write anything down! and he just talks to us like we're stupid if we ask questions." he crossed his arms over his chest, angrily furrowing his eyebrows.

"i can look in my dorm and see if i still have my notes from last semester, you can have them if i do."

"really?"

"i've already taken all the chemistry classes i need for my major, it's not like i have any use for them."

"so what happens until you find them?"

"you can't think of one thing your teacher talked about? or have any exam you've already taken, or something?"

he clicks a few times on his laptop, face scrunching with concentration as he scrolls.

"here's this." he turns the screen towards you, a triumphant smile on his face. a '35/100' is in large numbers next to the assignment name.

"damn. looks like i've got my work cut out for me."

"shut up!"

he searches through his surprisingly organized notebook, pulling out a few sheets of paper from a divider.

"this is the written work."

"give me a second." you look over the problems, analyzing his work and where he went wrong. his handwriting was a little wild, but to your surprise he was pretty detailed in writing out his calculations, which made it easier for you to explain his errors.

"so, you got all the numbers right on these but you have to mind your sig figs."

"what the fuck is a sig fig?"

"significant figures. in chemistry calculations, you have to be really accurate. sig figs are like a set of rules to keep numbers accurate."

"how is that any different from just using a calculator?"

you flip to an empty page in his notebook, making a small title to write about significant figures.

he watches as you scribble down the notes, your eyebrows creeping together with focus.

why is my chest tingling ?

ew this is weird

"so, every time you do a calculation you have to follow these rules with your answer."

"so, even though this is what the calculator says, that's not the whole answer."

"right, it starts with two zeros and those don't count."

it takes about a half an hour, but he's eventually able to get through a set of problems without error.

"good job."

"i did it! i'm the best chemistry student the world has ever seen!" he dramatically flexed his arms, proudly grinning

"you sure about that?"

"of course i am."

"okay, show me how to do dimensional analysis." his face pales, smile dropping. you let out a chuckle, turning to a fresh page in the notebook

"you couldn't just let me have my moment?"

"not until you don't need my help anymore."

two hours pass, and you've covered a good amount of information. you could tell as hard as he was trying, baji was quickly losing focus.

"okay, let's call it a night."

"finally." he breathed out, tossing his pencil on the table.

"damn, i didn't know you were that excited to get away from me."

"if you come near me with any more chemistry stuff right now, i might punch you."

"i'd like to see you try." you raise a brow, making him smirk.

"you’re a lot more dense than i thought, but i think you’ll be able to pass. the fact that you actually care so much helps.”

"don't call me dense, i don't know what it means." you burst out into laughter, making him roll his eyes before he lets out a chuckle.

"just because im in toman doesn't mean i don't care about school. we're all humans too, we don't eat sleep and breathe fighting."

"i never said you guys aren't human. but it's nice that you care about your grades."

"well don't make me out to be some softie!"

"i'll see you later baji." you roll your eyes, waving as you head for your dorm.

you open the door, hanging your keys on the wall hook.

"how was tutoring?"

you whirl around, seeing emma, draken, and mikey sitting on your floor with uno cards in their hands.

"how the hell did you guys get in here?!"

"picked the lock." mikey grinned

"you’re all insane."

"i'm surprised you both made it out alive." emma giggled

"he’s even dumber than i thought, but he’ll be fine.”

“baji might seem like an idiot, but he’ll surprise you.” draken doesn't look up from placing down a draw four, making mikey's jaw drop.

"kenny! this is betrayal!"

"you hit me with draw fours three times in a row last round, get over it."

you grin at their banter, mikey dramatically whining about draken breaking his heart.

"deal me in next round, i'll destroy you losers."

"you're on." emma grins, grabbing the stack of cards.

as the next round begins and you're looking over your cards, your phone buzzes.

unknown

today, 7:32 pm

hey

can we meet agn tmw

hey baji

more questions already?

i'm still at the lirbary

libary

LIBRARY

i was looking at the notes u rote and i got lost

same time tomorrow

[sig-fig-counter]

There are no significant figures in this post.

Just had a realization as I try to fall asleep:

I always held out that it's 'Sig Figs' because it's sorry for 'significant figures.' But it's Fig and the Cig Figs because she's being rebellious and so 'significant' is spelled with a 'c.'

It makes sense to me now.

[wiltcolton]

Best Significant Figures Calculator for Students

What are significant figures? Significant figures are all numbers that add up to the meaning of the overall value of the number. To avoid repeating figures that are not significant, numbers are often rounded. Care must be taken not to lose precision when rounding. Sometimes the goal of rounding numbers is simply to simplify them. Use the rounding calculator to solve these problems.

To Calculate you signifigant figures online to visit here.

 https://sigfig-calculator.com/

What are the rules for significant figures? To determine which numbers are significant and which are not, use the following rules:

to the left of the decimal value less than 1 is not significant. All trailing zeros that are placeholders are not significant. Zeros between nonzero numbers are significant. All non-zero numbers are significant. If a number has more numbers than the desired number of significant digits, the number is rounded. For example, 432,500 are 433,000 to 3 significant digits (using half a (regular) rounding) . Zeros at the end of numbers that are not significant but are not removed as removing it would affect the value of the number. In the example above, we can't remove 000 to 433,000 unless you change the number to scientific notation. How to use the sig-fig calculator Our significant numbers The calculator works in two modes: it performs arithmetic operations on multiple numbers (e.g. 4.18 / 2.33) or simply rounds a number to the desired number of Sig figs.

According to the rules given above, we can calculate sig-figs by hand or with.004562 and we want 2 significant figures. Trailing zeros are placeholders, so we don't count them. Next, we round 4562 to 2 digits, leaving us with 0.0046.

. Now we will consider an example that is not a decimal. Suppose we want 3,453,528 to 4 significant figures. We simply round the whole number to the nearest thousand, which gives us 3,454,000.

What if a number is in scientific notation? In such cases the same rules apply. , which replaces x 10 with an uppercase or lowercase letter 'e'. For example, the number 5,033 x 1023 is equivalent to 5,033E23 (or 5,033e23). For a very small number such as 6.674 x 10-11, the representation of the E notation is 6.674E-11 (or 6.674e-11) .

With estimation, the number of significant digits should not be greater than log base 10 of the sample size and rounded to the nearest integer. For example, if the sample size is 150, the logarithm of 150 is approximately 2. 18, so we use 2 significant figures

Significant figures in operations There are additional rules regarding operations: addition, subtraction, multiplication and division.

For addition and subtraction operations, the result must have no more decimal places than the number in the operation with For example, when performing the operation 128.1 + 1.72 + 0.457, the value with the least number of decimal places (1) is 128 By Therefore, the result must also have a decimal: 128.̲1 + 1.7̲2 + 0. 45̲7 = 130.̲277 = 130.̲3 The position of the last significant number is indicated by underlining

For multiplication and division operations, the result must have no more significant figures than the number of the operation with the least number of significant figures For example, When performing the operation 4.321 * 3.14, the value with the least significant figures (3) is 3.14, so the result must also be given to three significant figures: 4. 32̲1 * 3.1̲4 = 13.̲56974 = 13.̲6.

If only addition and subtraction are done, it is enough to do all the calculations at the same time and apply the rules of significant numbers to the final result.

If only multiplication and division are done, this is sufficient to do all the calculations at once and apply the rules of significant numbers to the final result.

But if you are doing mixed calculations - addition / subtraction and multiplication / division - you need to write down the number of significant numbers for each step of the calculation. For example, for the calculation 12.1̲3 + 1.7̲2 * 3. ̲4 After the first step, you will get the following result: 12.1̲3 + 5.̲848. Notice now that the result of the multiplication operation is accurate to 2 significant digits and, most importantly, to one decimal place. You shouldn't round down the subtotal and only apply the rules for significant digits to the final result. In this example, the final steps in the calculation are 12.1̲3 + 5.̲848 = 17.̲978 = 18.̲0.

Exact values, including defined numbers such as conversion factors and 'pure' numbers, do not affect the accuracy of the calculation. They can be treated as having an infinite number of significant numbers. For example when using the speed conversion: You have to multiply the value in m / s by 3.6 if you want to get the value in km / h. The number of significant numbers is further determined by the accuracy of the initial speed value in m /.

[badbackgroundscience]

Dr. Doom’s Doctorate isn’t in Anything that Requires SigFigs

Am I back from a many week-long hiatus? I might be back from a many week-long hiatus...

I first had to deal with significant figures in 10th grade. I hated them. I was the child who would write out every decimal value my calculator reported back to me, and sigfigs told me I couldn’t do that, anymore. I thought that I was being forced to be less accurate.

Several years later I ended up having to teach children how to use sigfigs, and it was only then that I actually grew to appreciate their significance.

It’s all about error, and the propagation of error.

Let me ask you a question: how tall are you? I’m roughly 164 cm. I say ‘roughly’ because 1. A human’s height varies over the course of the day as gravity compresses the spine, and 2. My ruler only goes down to millimeters, and while I could technically use it to make an estimate down to a tenth of a millimeter (e.g. I’m 164.02 cm), I know no human cares about me being that accurate.

So that 164 number carries with it some amount of inaccuracy. But if I use the height I normally say because I’m an American, 5″4.5″ (or 64.5 inches), I would have less inaccuracy. That’s because 1/10th of an inch is smaller than 1 centimeter.

If my measured height were 164.0, instead of just 164 with no tenths place, I’m suddenly 10 times more accurate. No longer am I somewhere between 163.5 and 164.4; I’m between 163.95 and 164.04. If I use the full capabilities of my ruler and get 164.02, I am really somewhere between 164.015 and 164.024.

But even down to a tenth of a millimeter, there’s some amount of uncertainty. It’s insignificant when recording your height at your next doctor’s visit, but not all measurements need the same level of accuracy.

For example, the seismometer run by the InSight lander is capable of measuring vibrations beneath the Martian surface that move the ground less than the diameter of a hydrogen atom.

That’s less than 0.0000000001 meters. A million times more accurate a measurement is needed than our pathetic tenth of a millimeter.

If the SEIS on InSight reports a movement of, say, 0.0117455253 meters, all those numbers (except the leading zeroes) matter. 

If the SEIS reports a movement of 0.0117400000 meters, all those zeroes after the non-zeroes matter, because the machine is that damn accurate. 

But if you suddenly do some math and combine that measurement with numbers that didn’t come from as accurate machine, you lose that accuracy. 

I remember having to do more complicated error propagation in an undergrad physics lab, but the stuff you learn in high school chemistry will serve most people’s purposes.

When adding/subtracting numbers with different levels of accuracy, you can only have an answer that is as accurate as your least accurate number. 

When multiplying/dividing numbers with different levels of accuracy, your answer can only have as many significant digits as your least accurate number.

(Yes, there is a slight difference, here. Don’t worry. I brought examples.

First, some sigfigs

5 -- 1 sigfig

5,000,000 -- also 1 sigfig

5,000,000. -- 7 sigfigs (note the decimal at the end - we never write numbers this way...)

5,000,001 -- 7 sigfigs

0.05 -- 1 sigfig

0.05000 -- 4 sigfigs

0.05001 -- 4 sigfigs

This might seem too complicated (”why do some zeroes matter but others don’t?”), but that’s why scientists write numbers in “scientific notation” which only reports significant numbers. Here are the same numbers written in that notation:

5 * 10^0

5 * 10^6

5.000000 * 10^6

5.000001 * 10^6

5 * 10^-2

5.000 * 10^-2

5.001 * 10^-2

It seems silly to write 50 as 5 * 10^1, but we do it because it could also have been 5.0 * 10^1. The first is a less accurate measurement (between 45 and 54) than the second (between 49.5 and 50.4).

Now for the math problem:

Pretend I’ve ordered you and three random strangers you’ve never met before to measure your collective height in meters -- I give each of you a ruler but they only have tick marks for imperial units.

You measure yourself and report 66 inches (That’s 2 sigfigs). The next person wants to be a little more accurate and says 69.5 inches (That’s 3 sigfigs). The third person just says 6 feet (1 sig fig). And the 4th person reports 60 inches (aka 6.0 * 10^1, not 6 * 10^1, so 2 sigfigs).

Your typical person would just add all this up (converting 6 feet to 72 inches) and get a total of 267.5 inches.

But that’s not how it works in sigfigs.

To convert 6 feet into inches, we do multiply by 12, but we can’t use 72. Our answer can only have 1 sigfig, so we have to round down to 70. That’s because only reporting 6 feet means, technically, there’s so much uncertainty he could be anywhere between 5.5 feet and 6.4 feet.* That puts the range of possible inches between 66 and 76.8 inches. Hence, we have to use 70 (i.e. 7 * 10^1) in.

So we add these numbers together, but sigfig rules requires we round our answer to the least accurate place value. For us, that’s the tens place because of that 70. So 265.5 gets rounded up to 270 (i.e. 2.7 * 10^2).

That’s the sum we have to use.

Had the third person told us 72 inches, we would get a sum of 267.5, and round to 268 inches.

Now we have to convert to meters. The typical conversion between imperial and metric units of distance is that 1 in = 2.54 centimeters (That’s 3 sigfigs). When we multiply, again, we can only as many sigfigs as our least accurate number, which is 2. 

270 in * 2.54 cm/in = 685.8 cm, 685.8 cm / 100 cm/m = 6.858 m

(100 cm is exactly, by definition, 1 meter. So even though 1 meter has only 1 non-zero number in it, and 3 sigfigs, we pretend we have an infinite number of sigfigs** when we divide by 100.)

The cumulative height of you and those three strangers is, with the correct number of significant figures 6.9 meters.

</mathproblem> You can all relax, now.

If you need a more accurate answer than that, then you need to take more accurate measurements. But not just one of them. Note that person who bothered to report their height to the nearest tenth of an inch - or if someone had measured to the nearest eighth of an inch - it wouldn’t have mattered because of Mr. I’m 6 feet tall over there.

* Based on anecdotal experience, he’s probably 5′11″.

** Infinite sigfigs also apply to when you’re counting things, like I have 12 wheels of cheese. Technically you can also have fractions of a wheel of cheese, so that muddies the waters a bit...

Fantastic Four Annual #2 - Stan Lee, Jack Kirby, Chic Stone

[trendsdresscom]

 How to count significant figures in a number?

Mathematics is a subject that most people are not fond of. It involves a lot of concept understanding and state of the art calculation abilities. It is not possible for every individual to possess these skills. This is the reason why people struggle when they do not have a liking for numbers, formulae and logic. The concepts related to significant figures are used for mathematics, chemistry and other disciplines. These calculations go well if you have a thorough understanding of the rules.

  Running through significant figures rules

Some of the key rules which you need to know about are listed below.

  All non-zero digits are counted as significant

 This is the basic but a highly important rule which is used when you are solving problems related to significant figures. Suppose that you have the following number.

2355 and you have to determine the number of significant digits in this number. In accordance with this rule, the answer is 4. The reason being that all digits are non-zero.

  In non-decimal numbers, zeros between two significant digits are counted as significant

Let us go through a proper example for understanding this point.

Suppose that you have the following number.

205500.

In the above number, one zero appears between 2 and 5 while the other two are present at the end. Hence, there will be a total of 4 significant figures. The last two zeroes would not be counted as they do not have significant digits on the left and right.

  The trailing zeros in decimal numbers are counted as significant

Consider that you have the number given below for which the significant digit count needs to be calculated.

0.20500

In the above number, there will be a total of 5 significant digits. As this number has a decimal, the last two zeroes would be counted as significant.

  Rules are not that simple to remember

For some people, it may not be that hard to remember these rules and implement them correctly.  It is all about how interested you are in mathematics. If you are not in love with the definitions and concepts, it would be quite hard for you to remember these rules and use them effectively. Unfortunately, most people do not have a serious liking for this discipline. Remembering rules for them is nothing less than climbing Mount Everest. An easier option for all such people is using a sig fig calculator.

  The Calculators.tech Sigfig Calculator

Using a tool and selecting a credible one are two very different things. Users face a lot of inconvenience when they end up with the wrong tool. Several alternatives for significant figure calculators are available on the internet. As a user, you have to be smart enough to pick the right one. The calculator by Calculators.tech is a must consideration since it is better than other tools developed for the same purpose.

  A strong programming framework to produce accurate outputs

The key purpose of using any tool is getting accurate results. Through manual methods, getting accurate results is not very much possible. Even the best individuals make errors while doing it. However, the accuracy of results in case of Sig fig Calculator also depends on the dependability of implemented algorithms.

The best programming practices have been used to develop this tool so accuracy is certainly not a problem. Once the tool has produced the results, there is no need to recheck anything. In other words, you can depend on the results produced without having any doubt in mind. It is not one of the calculators that get hung if the users use them multiple times.

   No need to be a mathematician

This calculator is not used by mathematicians only. Anyone who has to perform these calculations in one way or other would find this tool handy. The interface is quite easy so even if you do not in-depth knowledge of significant figures rules, no difference would be made.

At times, students working on chemistry assignments may come across significant figure problems. It is not necessary that they would be well aware of the rules and important points. Thus, using this calculator would be a better alternative for them.

  Free access is a plus

There is no need to swipe your credit card and make a digital purchase. The access to this tool is 100% free so the financial standing of the user does not matter at all. As no money has to be spent at all, any user can use it.

  What are the stages of using this calculator?

The following steps have to be completed once you have opened the link.

  Enter the number which you wish to round off

Consider that you want to round off the number 25366, simply enter it in the first box and move to the next step.

  Enter the number of significant digits for rounding off

How many digits do you want the number to be rounded off? Let us consider that in this case, you need to round off the number to 3 significant digits.

  Understanding / viewing the result

When you enter the inputs mentioned above, the following result will be shown on your screen.

25400

How was this generated? The right most digit is 6 which is greater than 5. Hence, 1 will be added to the digit on its left and the number would be dropped. Similarly, the second last digit is also a 6 so the same process will be repeated.

  Conclusion

This SigFig calculator is ideal for people who have weak concepts. It is not a piece of cake for everyone to learn these rules. A major percentage of people have no idea how these principles are applied. Hence, to solve these problems, an online calculator becomes a suitable alternative for such users.

This calculator does not put any accuracy problems on the table which is a big relief. In this way, none of the answers have to be rechecked and or confirmed.

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[letmestudyaflower]

 chem notes: MEASUREMENTS intro

CLOSE UPS: page 1, page 2

significant figures

common units + property

temp. conversation factors

power of ten conversions

common conversion factors

SEE MORE CHEM NOTES

type-able portion of my notes:

SIG FIG RULES

always significant if...

nonzero digits

captive zeros

trailing zeros(at the end AND in scientific notation)

never significant if...

leading zeros

trailing zeros

[instituteofreproducingme-blog]

ゼロ除算関係

№967 Dividing by Nothing　by Alberto Martinez Title page of Leonhard Euler, Vollständige Anleitung zur Algebra, Vol. 1 (edition of 1771, first published in 1770), and p. 34 from Article 83, where Euler explains why a number divided by zero gives infinity. https://notevenpast.org/dividing-nothing/　　より The Road Fig 5.2. Isaac Newton (1643-1727) and Gottfried Leibniz (1646-1716) were the culprits, ignoring the first commandment of mathematics not to divide by zero. But they hit gold, because what they mined in the process was the ideal circle. http://thethirty-ninesteps.com/page_5-the_road.php　より mercredi, juillet 06, 2011 0/0, la célèbre formule d'Evariste Galois ! http://divisionparzero.blogspot.jp/2011/07/00-la-celebre-formule-devariste-galois.html　　より 無限に関する様々な数学的概念：無限大 ：記号∞ （アーベルなどはこれを 1 / 0 のように表記していた）で表す。 大雑把に言えば、いかなる数よりも大きいさまを表すものであるが、より明確な意味付けは文脈により様々である。https://ja.wikipedia.org/wiki/%E7%84%A1%E9%99%90　より リーマン球面：無限遠点が、実は　原点と通じていた。 https://ja.wikipedia.org/wiki/%E3%83%AA%E3%83%BC%E3%83%9E%E3%83%B3%E7%90%83%E9%9D%A2　より http://jestingstock.com/indian-mathematician-brahmagupta-image.html　より ブラーマグプタ（Brahmagupta、598年 – 668年?）はインドの数学者・天文学者。ブラマグプタとも呼ばれる。その著作は、イスラーム世界やヨーロッパにインド数学や天文学を伝える役割を果たした。 628年に、総合的な数理天文書『ブラーマ・スプタ・シッダーンタ』（ब्राह्मस्फुटसिद्धान्त Brāhmasphuṭasiddhānta）を著した。この中の数章で数学が扱われており、第12章はガニタ（算術）、第18章はクッタカ（代数）にあてられている。クッタカという語は、もとは「粉々に砕く」という意味だったが、のちに係数の値を小さくしてゆく逐次過程の方法を意味するようになり、代数の中で不定解析を表すようになった。この書では、 0 と負の数にも触れていて、その算法は現代の考え方に近い（ただし 0 ÷ 0 ＝ 0 と定義している点は現代と異なっている） https://ja.wikipedia.org/wiki/%E3%83%96%E3%83%A9%E3%83%BC%E3%83%9E%E3%82%B0%E3%83%97%E3%82%BFより ブラーマ・スプタ・シッダーンタ (Brahmasphutasiddhanta) は、7世紀のインドの数学者・天文学者であるブラーマグプタの628年の著作である。表題は宇宙の始まりという意味。 数としての「0（ゼロ）の概念」がはっきりと書かれた、現存する最古の書物として有名である。https://ja.wikipedia.org/wiki/%E3%83%96%E3%83%A9%E3%83%BC%E3%83%9E%E3%83%BB%E3%82%B9%E3%83%97%E3%82%BF%E3%83%BB%E3%82%B7%E3%83%83%E3%83%80%E3%83%BC%E3%83%B3%E3%82%BF　より ゼロ除算の歴史��ゼロ除算はゼロで割ることを考えるであるが、アリストテレス以来問題とされ、ゼロの記録がインドで初めて６２８年になされているが、既にそのとき、正解1/0が期待されていたと言う。しかし、理論づけられず、その後１３００年を超えて、不可能である、あるいは無限、無限大、無限遠点とされてきたものである。 An Early Reference to Division by Zero C. B. Boyer http://www.fen.bilkent.edu.tr/~franz/M300/zero.pdf Impact of ‘Division by Zero’ in Einstein’s Static Universe and Newton’s Equations in Classical Mechanics：http://gsjournal.net/Science-Journals/Research%20Papers-Relativity%20Theory/Download/2084　より 神秘的に美しい3つの公式： 面白い事にゼロ除算については、いろいろな説が現在存在します しかし、間もなく決着がつくのではないでしょうか。 ゼロ除算は、なにもかも当たり前ではないでしょうか。 ラース・ヴァレリアン・アールフォルス（Lars Valerian Ahlfors、1907年4月18日-1996年10月11日）はフィンランドの数学者。リーマン面の研究と複素解析の教科書を書いたことで知られる。https://ja.wikipedia.org/wiki/%E3%83%A9%E3%83%BC%E3%82%B9%E3%83%BB%E3%83%B4%E3%82%A1%E3%83%AC%E3%83%AA%E3%82%A2%E3%83%B3%E3%83%BB%E3%82%A2%E3%83%BC%E3%83%AB%E3%83%95%E3%82%A9%E3%83%AB%E3%82%B9 フィールズ賞第一号 COMPLEX ANALYSIS, 3E (International Series in Pure and Applied Mathematics) (英語) ハードカバー – 1979/1/1 Lars Ahlfors (著) http://www.amazon.co.jp/COMPLEX-ANALYSIS-International-Applied-Mathematics/dp/0070006571/ref=sr_1_fkmr1_1?ie=UTF8&qid=1463478645&sr=8-1-fkmr1&keywords=Lars+Valerian+Ahlfors%E3%80%80%E3%80%80COMPLEX+ANALYSIS 原点の円に関する鏡像は、実は　原点であった。 本では、無限遠点と考えられていました。 Ramanujan says that answer for 0/0 is infinity. But I'm not sure it's ... https://www.quora.com/Ramanujan-says-that-answer-for-0-0-is-infi... You can see from the other answers, that from the concept of limits, 0/0 can approach any value, even infinity. ... So, let me take a system where division by zero is actually defined, that is, you can multiply or divide both sides of an equation by ... https://www.quora.com/Ramanujan-says-that-answer-for-0-0-is-infinity-But-Im-not-sure-its-correct-Can-anyone-help-me Abel　Memorial in Gjerstad Discussions: Early History of Division by Zero H. G. Romig The American Mathematical Monthly Vol. 31, No. 8 (Oct., 1924), pp. 387-389 Published by: Mathematical Association of America DOI: 10.2307/2298825 Stable URL: http://www.jstor.org/stable/2298825 Page Count: 3 ロピタルの定理 (ロピタルのていり、英: l'Hôpital's rule) とは、微分積分学において不定形 (en) の極限を微分を用いて求めるための定理である。綴りl'Hôpital / l'Hospital、カタカナ表記ロピタル / ホスピタルの揺れについてはギヨーム・ド・ロピタルの項を参照。ベルヌーイの定理 (英語: Bernoulli's rule) と呼ばれることもある。本定理を (しばしば複数回) 適用することにより、不定形の式を非不定形の式に変換し、その極限値を容易に求めることができる可能性がある。https://ja.wikipedia.org/wiki/%E3%83%AD%E3%83%94%E3%82%BF%E3%83%AB%E3%81%AE%E5%AE%9A%E7%90%86 Ein aufleuchtender Blitz: Niels Henrik Abel und seine Zeit https://books.google.co.jp/books?isbn=3642558402 - Arild Stubhaug - 2013 - ‎Mathematics Niels Henrik Abel und seine Zeit Arild Stubhaug. Abb. 19 a–c. a. ... Eine Kurve, die Abel studierte und dabei herausfand, wie sich der Umfang inn gleich große Teile aufteilen lässt. ... Beim Integralzeichen statt der liegenden ∞ den Bruch 1/0. https://books.google.co.jp/books?id=wTP1BQAAQBAJ&pg=PA282&lpg=PA282&dq=Niels+Henrik+Abel%E3%80%80%E3%80%80ARILD+Stubhaug%E3%80%80%E3%80%80%EF%BC%91/0%EF%BC%9D%E2%88%9E&source=bl&ots=wUaYL6x6lK&sig=OX1Yk_HxbCMm_FACotHYlgrbfsg&hl=ja&sa=X&ved=0ahUKEwj8-pftm-PPAhXIzVQKHX7ZCMEQ6AEISTAG#v=onepage&q=Niels%20Henrik%20Abel%E3%80%80%E3%80%80ARILD%20Stubhaug%E3%80%80%E3%80%80%EF%BC%91%2F0%EF%BC%9D%E2%88%9E&f=false Indeterminate: the hidden power of 0 divided by 0 2016/12/02 に公開 You've all been indoctrinated into accepting that you cannot divide by zero. Find out about the beautiful mathematics that results when you do it anyway in calculus. Featuring some of the most notorious "forbidden" expressions like 0/0 and 1^∞ as well as Apple's Siri and Sir Isaac Newton. https://www.youtube.com/watch?v=oc0M1o8tuPo　より ゼロ除算の論文： file:///C:/Users/saito%20saburo/Downloads/P1-Division.pdf　より Eulerのゼロ除算に関する想い： file:///C:/Users/saito%20saburo/Downloads/Y_1770_Euler_Elements%20of%20algebra%20traslated%201840%20l%20p%2059%20(1).pdf　より An Approach to Overcome Division by Zero in the Interval Gauss Algorithm http://link.springer.com/article/10.1023/A:1015565313636 Carolus Fridericus Gauss：https://www.slideshare.net/fgz08/gauss-elimination-4686597 Archimedes：Arbelos https://www.math.nyu.edu/~crorres/Archimedes/Stamps/stamps.html　より Archimedes Principle in Completely Submerged Balloons: Revisited Ajay Sharma： file:///C:/Users/saito%20saburo/Desktop/research_papers_mechanics___electrodynamics_science_journal_3499.pdf ［PDF]Indeterminate Form in the Equations of Archimedes, Newton and Einstein http://gsjournal.net/Science-Journals/Research%20Papers-Relativity%20Theory/Download/3222 このページを訳す 0. 0 . The reason is that in the case of Archimedes principle, equations became feasible in. 1935 after enunciation of the principle in 1685, when ... Although division by zero is not permitted, yet it smoothly follows from equations based upon. Thinking ahead of Archimedes, Newton and Einstein - The General ... gsjournal.net/Science-Journals/Communications.../5503 このページを訳す old Archimedes Principle, Newton' s law, Einstein 's mass energy equation. E=mc2 . .... filled in balloon becomes INDETERMINATE (0/0). It is not justified. If the generalized form Archimedes principle is used then we get exact volume V ..... http://gsjournal.net/Science-Journals/Communications-Mechanics%20/%20Electrodynamics/Download/5503 Find circles that are tangent to three given circles (Apollonius’ Problem) in C# http://csharphelper.com/blog/2016/09/find-circles-that-are-tangent-to-three-given-circles-apollonius-problem-in-c/　より ゼロ除算に関する詩： The reason we cannot devide by zero is simply axiomatic as Plato pointed out. http://mathhelpforum.com/algebra/223130-dividing-zero.html　より

[mathematicianadda]

Beach-ball like differential operators of a two-dimensional function

I'm looking for references, known names of, and other useful pointers and insight about (pairs of) differential operators that are "beach-ball like" because they sample a 2-dimensional function in these infinitesimally-sized regular patterns with the indicated alternating polarities:

Figure 1. Pairs of differential operators and a beach ball.

These include a hexagonal and a decagonal pattern. The pairs of operators can be formed by:

$$\begin{gather}\lim_{h\to 0}\frac{\sum_{N=0}^{4N + 1} (-1)^n f\bigg(x + h\cos\left(\frac{2\pi n}{4N + 2}\right), y + h\sin\left(\frac{2\pi n}{4N + 2}\right)\bigg)}{h^{2N + 1}},\\ \lim_{h\to 0}\frac{\sum_{N=0}^{4N + 1} (-1)^n f\bigg(x + h\sin\left(\frac{2\pi n}{4N + 2}\right), y + h\cos\left(\frac{2\pi n}{4N + 2}\right)\bigg)}{h^{2N + 1}},\end{gather}\tag{1}$$

although I'm not certain about the normalization factor $h^{-(2N+1)}$, which at least does not collapse the operator to zero or blow it up to infinity for $N=0$, which is simply a coefficient times differentiation:

$$\begin{gather}N=0:\\ 2\frac{d}{dx}f(x, y),\\ 2\frac{d}{dy}f(x, y),\end{gather}\tag{2}$$

or for $N=1$, which I think is:

$$\begin{gather}N=1:\\ \frac{1}{4}\left(\frac{d}{dx}\right)^3f(x,y)-\frac{3}{4}\frac{d}{dx}\left(\frac{d}{dy}\right)^2f(x, y),\\ \frac{1}{4}\left(\frac{d}{dy}\right)^3f(x,y)-\frac{3}{4}\frac{d}{dy}\left(\frac{d}{dx}\right)^2f(x, y).\end{gather}\tag{3}$$

Applying these to a 2-d Gaussian function and plotting:

Figure 2. Color-mapped 1:1 scale (pixel:unit) plots of, in order: A 2-d Gaussian function with standard deviation $\sigma = 16$, derivative of the Gaussian function with respect to horizontal coordinate $x$, differential operator $\frac{1}{4}\big(\frac{d}{dx}\big)^3-\frac{3}{4}\frac{d}{dx}\big(\frac{d}{dy}\big)^2$ applied to the Gaussian function. Color key: blue: minimum, white: zero, red: maximum.

Python source for Fig. 2:

import matplotlib.pyplot as plt import numpy as np import scipy.ndimage sig = 16 # Standard deviation N = 161 # Image width x = np.zeros([N, N]) x[N//2, N//2] = 1 h = scipy.ndimage.gaussian_filter(x, sigma=[sig, sig], order=[0, 0], truncate=(N//2)/sig) ddx = scipy.ndimage.gaussian_filter(x, sigma=[sig, sig], order=[0, 1], truncate=(N//2)/sig) h1x = scipy.ndimage.gaussian_filter(x, sigma=[sig, sig], order=[0, 3], truncate=(N//2)/sig) - 3*scipy.ndimage.gaussian_filter(x, sigma=[sig, sig], order=[2, 1], truncate=(N//2)/sig) plt.imsave('h.png', plt.cm.bwr(plt.Normalize(vmin=-h.max(), vmax=h.max())(h))) plt.imsave('ddx.png', plt.cm.bwr(plt.Normalize(vmin=-ddx.max(), vmax=ddx.max())(ddx))) plt.imsave('h1x.png', plt.cm.bwr(plt.Normalize(vmin=-h1x.max(), vmax=h1x.max())(h1x))) plt.imsave('gaussiankey.png', plt.cm.bwr(np.repeat([(np.arange(N)/(N-1))], 16, 0)))

from Hot Weekly Questions - Mathematics Stack Exchange from Blogger http://bit.ly/2wfuMeZ

[bonniejstarks]

 How to count significant figures in a number?

Mathematics is a subject that most people are not fond of. It involves a lot of concept understanding and state of the art calculation abilities. It is not possible for every individual to possess these skills. This is the reason why people struggle when they do not have a liking for numbers, formulae and logic. The concepts related to significant figures are used for mathematics, chemistry and other disciplines. These calculations go well if you have a thorough understanding of the rules.

Running through significant figures rules

Some of the key rules which you need to know about are listed below.

All non-zero digits are counted as significant

This is the basic but a highly important rule which is used when you are solving problems related to significant figures. Suppose that you have the following number.

2355 and you have to determine the number of significant digits in this number. In accordance with this rule, the answer is 4. The reason being that all digits are non-zero.

In non-decimal numbers, zeros between two significant digits are counted as significant

Let us go through a proper example for understanding this point.

Suppose that you have the following number.

205500.

In the above number, one zero appears between 2 and 5 while the other two are present at the end. Hence, there will be a total of 4 significant figures. The last two zeroes would not be counted as they do not have significant digits on the left and right.

The trailing zeros in decimal numbers are counted as significant

Consider that you have the number given below for which the significant digit count needs to be calculated.

0.20500

In the above number, there will be a total of 5 significant digits. As this number has a decimal, the last two zeroes would be counted as significant.

Rules are not that simple to remember

For some people, it may not be that hard to remember these rules and implement them correctly.  It is all about how interested you are in mathematics. If you are not in love with the definitions and concepts, it would be quite hard for you to remember these rules and use them effectively. Unfortunately, most people do not have a serious liking for this discipline. Remembering rules for them is nothing less than climbing Mount Everest. An easier option for all such people is using a sig fig calculator.

The Calculators.tech Sigfig Calculator

Using a tool and selecting a credible one are two very different things. Users face a lot of inconvenience when they end up with the wrong tool. Several alternatives for significant figure calculators are available on the internet. As a user, you have to be smart enough to pick the right one. The calculator by Calculators.tech is a must consideration since it is better than other tools developed for the same purpose.

A strong programming framework to produce accurate outputs

The key purpose of using any tool is getting accurate results. Through manual methods, getting accurate results is not very much possible. Even the best individuals make errors while doing it. However, the accuracy of results in case of Sig fig Calculator also depends on the dependability of implemented algorithms.

The best programming practices have been used to develop this tool so accuracy is certainly not a problem. Once the tool has produced the results, there is no need to recheck anything. In other words, you can depend on the results produced without having any doubt in mind. It is not one of the calculators that get hung if the users use them multiple times.

 No need to be a mathematician

This calculator is not used by mathematicians only. Anyone who has to perform these calculations in one way or other would find this tool handy. The interface is quite easy so even if you do not in-depth knowledge of significant figures rules, no difference would be made.

At times, students working on chemistry assignments may come across significant figure problems. It is not necessary that they would be well aware of the rules and important points. Thus, using this calculator would be a better alternative for them.

Free access is a plus

There is no need to swipe your credit card and make a digital purchase. The access to this tool is 100% free so the financial standing of the user does not matter at all. As no money has to be spent at all, any user can use it.

What are the stages of using this calculator?

The following steps have to be completed once you have opened the link.

Enter the number which you wish to round off

Consider that you want to round off the number 25366, simply enter it in the first box and move to the next step.

Enter the number of significant digits for rounding off

How many digits do you want the number to be rounded off? Let us consider that in this case, you need to round off the number to 3 significant digits.

Understanding / viewing the result

When you enter the inputs mentioned above, the following result will be shown on your screen.

25400

How was this generated? The right most digit is 6 which is greater than 5. Hence, 1 will be added to the digit on its left and the number would be dropped. Similarly, the second last digit is also a 6 so the same process will be repeated.

Conclusion

This SigFig calculator is ideal for people who have weak concepts. It is not a piece of cake for everyone to learn these rules. A major percentage of people have no idea how these principles are applied. Hence, to solve these problems, an online calculator becomes a suitable alternative for such users.

This calculator does not put any accuracy problems on the table which is a big relief. In this way, none of the answers have to be rechecked and or confirmed.

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[oflgtfol]

i hate college physics because the definition of sig figs is so different from what i learned in high school i literally nevr fucking know

THAT IS 3 SIG FIGS!!!!!!!! sure i probably calculated it wrong but either way dont tell me i didnt use 3 sig figs when i did!!!! zeroes after a decimal place count as sig figs!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!