Mental arithmetic comprises arithmetical calculations using only the human brain, with no help from calculators, computers, or pen and paper.

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Soroban Abacus - how to memorise?

I saw a video where Indian kids where fiddling fingers in the air and working out big sums. WOW I thought. I researched loads and have figured out that it's all based on using the Soroban Abacus (a ...
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Is there a speedy mental algorithm for subtracting large numbers?

Like for example 65465-78954-12356 = -25845 Obviously the "borrowing" method that everyone learns in elementary school works, but it's slow and tedious, especially for results that come out negative. ...
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Mental Primality Testing

At a trivia night, the following question was posed: "What is the smallest 5 digit prime?" Teams (of 4) were given about a minute to write down their answer to the question. Obviously, the answer is ...
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Are there human integrators? [closed]

Are there human "integrators", "differentiators", or "analysts"? I've heard of and seen people capable of performing seemingly complex arithmetic calculations mentally. Often this involves memorizing ...
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Toughest sequence's next term which is still unresolved

I went through a exam and the following question where I was stuck please tell me how to solve such types of question. Could anybody please answer me with full explanation. Thanks in advance. 1. $7, ...
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Is there a known simple mental approximation to a hypergeometric distribution?

Reading this question about calculating approximate lottery odds inspired me to ask about approximating the values of a hypergeometric distribution. Specifically, the situation I often find myself in ...
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Calculating power without using a calculator, for example $1.05^{10}$

How to find (or estimate) $1.05^{10}$ without using a calculator? Do we have any fast algorithm for cases where base is slightly more than one? Say up to $1.1$ with tick $0.05$.
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1answer
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Best algo for finding no. of steps required to convert a sequence to a palindromic sequence

[My first question of Math SE, so, HI!] I'm not sure of what the rules are around the place, but I have a straightforward question as follows... The sequences 23, 45, 23 and 23, 45, 56, 23, 23, ...
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fast mental arithmetic: is it an algorithm or table-like structure?

edit Removing the fluff, the question is: When solving problem X by heart, how does the mind reaches the solution very fast? by 'running' an algorithm or 'accessing' a table The 'fluffy' version: ...
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Why is Division harder than Multiplication?

Both conceptually and computationally it feels easier to see that: $ 6 \cdot 3.7 = 22.2$ than it is to see that $ 22.2 \div 6 = 3.7 $. Thoughts about the roots of this asymmetry? An analogous ...
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Why is multiplication commutative - intuitive explanation [duplicate]

While I know that both addition and multiplication are commutative operations, I can easily visualize that, e.g. 3 + 4 = 4 + 3 = 7 by thinking of seven objects in a row and separating them into two ...
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106 views

What is the value of $(72^2 - 64^2) : (44^2 - 24^2)$ [closed]

What is the value of $(72^2 - 64^2) : (44^2 - 24^2)$ How to calculate this without calculator?
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Find the numbers; Arithmetic Progression.

The sum of four consecutive numbers in an A.P is $28$. The product of the second and third numbers exceeds that of the first and last by $18$. Find the numbers. I thought of this: $$S_{4} = ...
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892 views

Better Divisibility by 8

Everywhere I look, when you want to see if something is divisible by $8$ then you see if the last $3$ digits are divisible by eight. But how do you know if the last $3$ digits are divisible by $8$? ...
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2answers
156 views

mental math: approximating fractional exponents

Does anyone have any good tricks for estimating expressions with fractional exponents (besides guess and check)? For example, I want to easily calculate $9.1^{1/3}$. Currently, the best I've got is ...
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Mental Math Techniques [closed]

What are some interesting mental math techniques that you know? Here's one that I got from my Grandmother who got it from a book: To square a two-digit number (from $26$ to $49$), take the number ...
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349 views

Fastest way to multiply small numbers with decimals mentally

Is there a fast way to multiply these numbers mentally? Example 1: 0.85 * 1,15 Example 2: 0.5 * 1.5 Example 3: 0.2 * 1,4
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The optimal way of improving fluency in elementary maths techniques, whilst holding a 9-5 job?

As an example of someone who has discovered maths at a later point in my life than average, and who has (perhaps unusually?) proven the point that it is perfectly possible to study undergraduate ...
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Remembering numbers while performing calculations mentally

Specific question: how can I improve on mentally performing calculations like these : $$(78-59)\times23-8\%\times(270+130)$$ ...
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2answers
224 views

Fastest way to multiply numbers mentally?

I'm wondering what is the fastest way to multiply numbers? For now, let's focus on 2-digit numbers and were one cannot use scrap paper. I've come across 3 fast methods: 64x43 1) ...
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1answer
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Flies caught by 100 spiders in 100 minutes

I went through the following question: If 5 spiders can catch five flies in five minutes. How many flies can hundred spiders catch in 100 minutes? The answer is calculated by the following ...
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2answers
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Understanding how This Works (shortcut for squaring $2$ digit numbers mentally)

We all know that math is as much about finding the answer as it is about knowing how a method leads you to the answer. In fact, not knowing the how has caused me to loose marks on several occasions ...
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Mental Math: Finding Square Roots to 1 Decimal Point

I have 2 questions here. What is the most effective and easy way of calculating square roots in your head to an accuracy of 1 decimal point? This would need to work with at least two digit, ...
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Mental estimate for tangent of an angle (from $0$ to $90$ degrees)

Does anyone know of a way to estimate the tangent of an angle in their head? Accuracy is not critically important, but within $5%$ percent would probably be good, 10% may be acceptable. I can ...
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335 views

Faster mental arithmetic with powers of 10

Please excuse me if this question is too vanilla. What's a faster way to do mental arithmetic involving powers of ten? I've always had to do this and I do it using scientific notation which I'm ...
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How to factor 5671?

The other day I wanted to factor 5671 in my head. (It turns out to be $53\cdot107$, but I did not know this at the time.) I quickly ruled out the easy divisors, 2, 3, 5, 7, 11, and 13. At this point ...
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Mental Math - Estimating Logarithms

How can we estimate logarithms with different bases? Take $\log_2 10$ ($1\over\log_{10}2$$\approx3.32192809$) for example. If we convert $10$ to binary, we get $1010_2$. So $\log_21010_2$ can clearly ...
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Numeric synaesthesia: uses of and advice for learning math.

It turns out that my adolescent son might have numeric synaesthesia-- numbers have specific colors and possibly other distinguishing characteristics for him. He has shown that he can commit long ...
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168 views

Show that $\left(\frac{1}{2}\left(x+\frac{2}{x}\right)\right)^2 > 2$ if $x^2 > 2$

Okay, I'm really sick and tired of this problem. Have been at it for an hour now and we all know the drill: if you don't get to the solution of a simple problem, you won't, so ... I'm working on a ...
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482 views

How to obtain an approximate expression for $\sqrt{\varepsilon}$ where $\varepsilon \ll 1$?

Is there a way to obtain an approximate expression for the square root $\sqrt{\varepsilon}$ of a small number $\varepsilon \ll 1$? To be more precise, I would like to have an expression which (1) I ...
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295 views

mental maths 9C4, 6^5 etc. [closed]

How can I calculate the following things in my head? $_9C_4$ I know this is $\frac{9\cdot 8\cdot 7\cdot 6}{4\cdot 3\cdot 2\cdot 1}$ and then $3\cdot 2\cdot 3\cdot 7$ but I can't immediately come up ...
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$f(n)=7^{6n} - 6^{6n}$, where $n$ is a positive integer.

$f(n)=7^{6n} - 6^{6n}$, where $n$ is a positive integer. find the divisors of $f(n)$ for odd and even values of $n$. Is there a general solution for the divisors. $$f(1)=7^6-6^6=(7^3)^{2}-(6^3)^{2}$$ ...
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Square three digit numbers, the efficient way

I would like to square a three digit number in my head. Now I know that the formula is $$ ( X + r ) ( X - r ) + r^2 = X^2 - rX + rX - r^2 + r^2 = X^2 $$ Where $\,r\,$ is a number such that $\,X + ...
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a big number that is obviously prime?

I once heard it asserted that $91$ is the smallest composite number that is not obviously composite. The reasoning was that any composite number divisible by $2$, $3$, or $5$ is obviously composite, ...
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372 views

Fast arithmetic, without a calculator?

This has been on my mind for quite a while now... Is it really crucial to be able to crunch numbers on the fly? I have considerate difficulty making out the quotient of $1 / 0.732 $ for example. I ...
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Books or site/guides about calculations by hand and mental tricks?

Any ideas about books I can get, from amazon? I need to get really good at mental math and math by hand because I'm taking an exam soon and that without a calculator. Thanks.