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

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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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3answers
188 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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0answers
29 views

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
27 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
26 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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3answers
141 views

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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109 views

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
213 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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3answers
417 views

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
135 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
61 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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4answers
4k 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
46 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, ...
2
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3answers
358 views

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: ...
6
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3answers
171 views

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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0answers
38 views

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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2answers
25 views

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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7answers
1k 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 $8$? ...
3
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2answers
301 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 ...
4
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3answers
402 views

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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3answers
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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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0answers
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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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0answers
106 views

Remembering numbers while performing calculations mentally

Specific question: how can I improve on mentally performing calculations like these : $$(78-59)\times23-8\%\times(270+130)$$ ...
3
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2answers
877 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
209 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 ...
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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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2answers
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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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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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2answers
528 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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3answers
252 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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2answers
645 views

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
178 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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3answers
175 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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2answers
643 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
148 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. $$f(1)=7^6-6^6=(7^3)^{2}-(6^3)^{2}$$ ...
2
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3answers
4k views

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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5answers
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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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10answers
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What is the fastest way to multiply two digit numbers?

I been playing different math games on my Android lately (for example: Math Cruncher). I've noticed that I'm unable to quickly (under 7-8 seconds) multiply two digit numbers (i.e $ 18 * 17$). So my ...
2
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2answers
470 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 ...
6
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7answers
636 views

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.