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

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Simplifying this fraction in a different base

Note: I would appreciate a solution that DOES NOT convert back to base 10. How would one simplify $\frac{43}{70}_8$? I assume, like in decimal, I must recognize a common factor and divide by that ...
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1answer
44 views

Faster mental calculations

Currently I am preparing for trading exams which tests faster arithmetic skills. For ex- 80 questions (like calculating 0.abc* 0.cd) to be done in 8 minutes. I am trying to memorize the squares and ...
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3answers
93 views

Shortcut for finding cube of the Numbers

Is there a shortcut for finding cube of a particular number like $68^3$ ? If anyone knows how to solve for two- and three digit numbers, can you please share the answer?
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1answer
83 views

Nice approximation to pattern

Does anyone know a good general approximation (within 5 percent) for the sum of the products of two numbers such that the sum of one number of one product and another of another product is equal two ...
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5answers
463 views

What is the fastest method to find which of $\frac {3\sqrt {3}-4}{7-2\sqrt {3}} $ and $\frac {3\sqrt {3}-8}{1-2\sqrt {3}} $ is bigger manually?

What is the fastest method to find which number is bigger manually? $\frac {3\sqrt {3}-4}{7-2\sqrt {3}} $ or $\frac {3\sqrt {3}-8}{1-2\sqrt {3}} $
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1answer
47 views

Is there a simple way of computing when $a^n=b^m$

I don't want exact equality just close enough to be useful in approximation. i.e. $2^{10 }= 10^3$ is very useful and used daily for an approximation. Is there a do this efficiently? Is there a way ...
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339 views

Did Feynman mentally compute $\sqrt[3]{1729.03}$ by linear approximation?

In the biopic ``infinity'' about Feynman. (11:48~15:50) Feynman compute $\sqrt[3]{1729.03}$ by a mental calculation. I guess that he use the linear approximation. That is, he observe that $1728=12^3$....
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1answer
72 views

How can I mentally calculate $cos(x), x∈(0.7, 1.2)$

I'm trying to learn how to calculate trig functions in my head. I'm planning on learning $\cos(x), x∈[0,π/2]$ and then using symmetry to calculate the others. I think the quadratic Maclaurin series ...
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3answers
166 views

Calculating trigonometric function values mentally

This may sound dumb, but does such a way exist to mentally (and quickly) determine the values of trigonometric functions such as $\sin(47^\circ)$ and so forth--quickly being a mere matter of seconds? ...
4
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1answer
138 views

Mental multiplication of two digit numbers

There is a common way of mentally doing squares of two-digit numbers all the way up to 19*. This is how it works (example is for calculating $13^2 = 13*13 = xyz$): The ones digit is the ones digit ...
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283 views

Calculating logs in your head

I quite often need to estimate $\log_2(x)$ for positive integer values of $x$. I find it frustrating that I have to rely on a calculator/computer to do this. By way of contrast, I can happily ...
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Finding the percent of a division fast and mentally

3/8= (0.125*3) = 0.375 = 37.5% is easy to calculate mentally but is there a better way to find the percent of the following divisions fast and mentally? 3.5/8 4.5/7
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Does anyone know of a Mental Math Game for blind students?

I'm looking for any computer game made for blind students where the math exercises are asked through a computer's speaker and answered through a microphone by the student. Unlike normal math games, ...
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0answers
29 views

How to develop number sense while finding out the middle/median value?

I have seen that many people can find out the middle value between numbers orally. My question is how to find the middle value orally? For example , the middle value between - 24576.5 MHz — 24881.5 ...
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19answers
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Mental Calculations

This is the famous picture "Mental Arithmetic. In the Public School of S. Rachinsky." by the Russian artist Nikolay Bogdanov-Belsky. The problem presented on a blackboard requires computing the ...
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2answers
28 views

How to estimate magnitude of expontent?

When given an exponent, such as 6^12, is there a simple way to approximate how large(magnitude) the result is, without performing the calculation? Is this method accurate for large exponents?
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mental ability math question [closed]

A worker may claim Rs. 15 for each km if he travels by taxi and Rs. 5 for each km if he drives his own car. If in one week he claimed Rs. 500 for travelling 80 km, how many kms did he travel by taxi ?
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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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3answers
233 views

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 ...
3
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0answers
147 views

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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1answer
76 views

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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5answers
8k views

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
52 views

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, 56,...
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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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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} = \frac{4}{...
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6k views

Daily exercises to speed up my mental calculations?

When I was a kid in school my father prevented me from using a calculator when solving my math homeworks. However at that time I was not convinced as of why not to use such a useful tool! So I kept on ...
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7answers
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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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396 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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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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118 views

Remembering numbers while performing calculations mentally

Specific question: how can I improve on mentally performing calculations like these : $$(78-59)\times23-8\%\times(270+130)$$ $$[2\text{digit}+-2\text{digit}]\times2\text{digit}+-x\%\text{of}[3\text{...
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2answers
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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
285 views

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 situation:...
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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, non-...
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8answers
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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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591 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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4answers
352 views

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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2answers
180 views

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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177 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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727 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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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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2answers
152 views

$f(n)=7^{6n} - 6^{6n}$, 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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3answers
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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 + r\,...
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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, ...