Mathematical Shortcuts
To find a square of a number:
Find the square number before it. Take the number to square and multiply it by two, and subtract one. Add the number times two minus one to the previous square number, and you have your answer. i.e. 26 squared is 25 squared is 625 plus 51, which is 676.
Proof: By the binomial theorem, (a + b)2 = a2 + 2ab + b2. b = 1 here, so we have (a + 1)^2 = a^2 + 2a + 1.
or,
If you have a long number to square, i.e. 6024, square 24, to get 576, square six, and multiply the answer by a million, to get 36,000,000, and multiply 6 by twenty-four by two by 1,000, to get 288,000, and add them all. The answer is 36,288,576.
No proof needed. This is just a simpler way of multiplying.
or, the Gilbreth method of squaring:
Take the number to be squared. I'll use 67. Take 67 - 25. The result is 42. Multiply by 100. Take 67-50. The result is 13. Square it. The answer is 169. Add 4200 + 169. The result is 4369. (Check this on a calculator if you are not sure.) If you would like to square a number between 25 and 50, take n-25, let's use 37. You get 12. Multiply by 100. Add 13 squared. 1369. If you want to square a number less than 25, a. MEMORIZE, as I have done. b. Let's use 23. 23-25 is -2. Multiply by 100. -200. 27 squared is 729. (Know it. It's 3^6, too.) 729 - 200 is 529.
Proof: 100(n – 25) + (50 – n)2 = 100n – 2500 + 2500 – 100n + n2 = n2 by distribution and the binomial theorem.
To multiply two even numbers or two odd numbers:
Average the two numbers. Square the average. Find the difference between the average and the lower number (or the higher number, it doesn't matter). Square the difference. Take the average squared and subtract the difference squared. i.e. 13 * 7. Average = 10, 102=100. 32=9. 100-9=91.
Proof: 
Carl Gauss method of consecutive addition:
. i.e. 1 - 50. Add 1 + 50. 51*50/2=1275.
To multiply a number and eleven
Take the two-digit number and put a 0 in the middle of the two digits. Add together the digits. Multiply that number by ten, and add it to the number. i.e. 48*11= [408+(12*10)]=528.
Proof: 11n = n + 10n, so each number can be added to its left neighbor.
Or
Take a number (let's take 257). Bring down the number in the one's place (7). Next we add the tens digit and the ones digit. We get 12, for a total of 127. Next we bring down the sum of the hundreds digit and the tens digit. We get 7, for a total of 827. Next we bring down the hundreds digit for a total of 2857.
To do subtraction without borrowing. I credit this
method to my friend, Aaron Lehmann
Set the problem up the normal way and do the subtraction of one column. If it is a negative number, write it as a negative number, continue doing this. At the end, for all the negative numbers, take one out of the digit higher, and do ten minus the negative number.
THE ULTIMATE MATHEMATICAL SHORTCUT: Newton’s method of finding the square root of a number without cut and try.
For example, let’s find the square root of 7 to five digits. First we guess, perhaps, 3. x1 = 3 – (32 – 7)/6 = 2.66667. x2 = 2.66667 – (2.666672 – 7)/5.33334 = 2.64583. x3 = 2.64583 – (2.645832 – 7)/5.29166 = 2.64575. x4 = 2.64575 – (2.645752 – 7)/5.29150 = 2.64575. Since two consecutive results match, we are done.
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