The secret of mental math

Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks by Arthur T. Benjamin

Originally published: 2006

Summarize the method in three words: Left to right.

Since most human memory can hold only about seven or eight digits at a time, this is about as large a problem as you can handle without resorting to artificial memory devices, like fingers, calculators, or the mnemonics taught in Chapter 7.

Gauss’s desire to better understand Nature through the language of mathematics was summed up in his motto, taken from Shakespeare’s King Lear (substituting “laws” for “law”):“Thou, nature, art my goddess; to thy laws/My services are bound.”  (p20)

Chapter 0 Quick Tricks: Easy (and Impressive) Calculations

32 × 11

To solve this problem, simply add the digits, 3+2 = 5, put the 5 between the 3 and the 2, and there is your answer: 352  (p1)

314 × 11, 

the answer still begins with 3 and ends with 4. Since 3+1= 4, and 1+4=5, the answer is 3454. (p4)

• Square numbers: 

1. The answer begins by multiplying the first digit by the next higher digit.

2. The answer ends in 25.

square the number 35, we simply multiply the first digit (3) by the next higher digit (4), then attach 25. Since 3×4=12, the answer is 1225. Therefore, 35×35=1225. (p5)

We can use a similar trick when multiplying two-digit numbers with the same first digit, and second digits that sum to 10. The answer begins the same way that it did before (the first digit multiplied by the next higher digit), followed by the product of the second digits. For example, let’s try 83× 87. (Both numbers begin with 8, and the last digits sum to 3+7 =10.) Since 8 ×9 =72, and 3×7= 21, the answer is 7221. (p5)

Chapter 1 A Little Give and Take:.Mental Addition and Subtraction

there are many good reasons why it is better to work from left to right. After all, you read numbers from left to right, you pronounce numbers from left to right, and so it’s just more natural to think about (and calculate) numbers from left to right. When you compute the answer from right to left (as you probably do on paper), you generate the answer backward. That’s what makes it so hard to do math in your head. Also, if you want to estimate your answer, it’s more important to know that your answer is “a little over 1200” than to know that your answer “ends in 8.” Thus, by working from left to right, you begin with the most significant digits of your problem. If you are used to working from right to left on paper, it may seem unnatural to work with numbers from left to right. But with practice you will find that it is the most natural and efficient way to do mental calculations.

It also illustrates a fundamental principle of mental arithmetic—namely, to simplify your problem by breaking it into smaller, more manageable parts. This is the key to virtually every method you will learn in this book. To paraphrase an old saying, there are three components to success—simplify, simplify, simplify.

• Subtraction:

1. First subtract 20, then subtract 9:

86 - 29 = 66 - 9 = 57 (first subtract 20) (then subtract 9)

But for this problem, I would prefer the following strategy:

2. First subtract 30, then add back 1:

86 - 29 = 56  + 1 =  57 (first subtract 30) (then add 1)

Here is the rule for deciding which method to use: If a two-digit subtraction problem would require borrowing, then round the second number up (to a multiple of ten). Subtract the rounded number, then add back the difference.  (p22)

• Using Complements 

Quick, how far from 100 are each of these numbers?

57 68 49 21 79

Here are the answers:

Notice that for each pair of numbers that add to 100, the first digits (on the left) add to 9 and the last digits (on the right) add to 10. We say that 43 is the complement of 57, 32 is the complement of 68, and so on. 

the complements are determined from left to right. As we have seen, the first digits add to 9, and the second digits add to 10. (An exception occurs in numbers ending in 0—e.g., 30 + 70 =100)

Chapter 2 Products of a Misspent Youth: Basic Multiplication  (1:10:35)