By Arthur Wouk

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**Extra info for A course of applied functional analysis**

**Example text**

Yes, 7 + 2 = 9, so we cross them out. We add the other digits, 5 + 7 + 8 = 20. And 2 + 0 = 2. Two is our substitute answer. I write the substitute numbers in pencil above or below the actual numbers in the problem. It might look like this: Did we get the right answer? We multiply the substitute numbers, 5 times 4 equals 20, which equals 2 (2 + 0 = 2). This is the same as our substitute answer, so we were right. Let’s try one more example: 456 × 831 = 368 936 We write in our substitute numbers: That was easy because we cast out (or crossed out) 4 and 5 from the first number, leaving 6; we crossed out 8 and 1 from the second number, leaving 3; and almost every digit was crossed out of the answer, 3 and 6 twice, and a 9.

It is a matter of personal preference. Simply choose the reference number you find easier to work with. Numbers above and below 20 The third possibility is if one number is above and the other below 20. For example: We can either add 18 to 12 or subtract 2 from 32, and then multiply the result by our reference number: 32 − 2 = 30 30 × 20 = 600 We now multiply the numbers in the circles: 2 × 12 = 24 It is actually minus 2 times 12 so our answer is −24. ) Let’s check the answer by casting out the nines: Zero times 5 is 0, so the answer is correct.

So, how do we solve the problem? 0? 0. The first number, ‘8’, signifies that the number equals eight. 0’, signifies the number equals eight and that it is accurate to one decimal place. The decimal point doesn’t change the value. So, here we go: Now, the problem is easy. Subtract diagonally. 68 − 20 = 48 Multiply 48 by the reference number (100) to get 4800. Multiply the numbers in the circles. ) 4800 + 640 = 5440 Thus: Now, we have to place the decimal point. How many digits are there after the decimal point in the problem?