By Alvin K Bettinger; John A Englund
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Additional info for Algebra and trigonometry
Calculate 16 × 16 using 10 as a reference: How about 512 × 512? 524 times 500 is 524 times 1000 divided by 2. 524 × 1000 = 524 000 or 524 thousand. Half of 524 thousand is 262 000. You could split up the 524 thousand into 500 thousand and 24 thousand. Both are easy to halve mentally. Half of 500 thousand is 250 thousand. Half of 24 thousand is 12 thousand. 250 thousand plus 12 thousand is 262 thousand. Now multiply the numbers in the circles: 12 × 12 = 144 262 000 + 144 = 262 144 (answer) Multiplying lower numbers Let’s try 6 × 4: We use a reference number of 10.
Can we make our method or formula work in this case? We can, but we must use a different reference number. Let’s try a reference number of 5. Five is 10 divided by 2, or half of 10. The easy way to multiply by 5 is to multiply by 10 and halve the answer. Six is above 5 so we put the circle above. Four is below 5 so we put the circle below. Six is 1 higher than 5 and 4 is 1 lower, so we put 1 in each circle. We add or subtract diagonally: 6 − 1 = 5 or 4 + 1 = 5 We multiply 5 by the reference number, which is also 5.
Did we get it right this time? The 936 cancels out, so we add 3 + 7 + 8, giving us 18, which adds up to 9, which in turn cancels, leaving 0. This is the same as our check answer, so this time we have it right. Does casting out nines prove we have the right answer? No, but we can be almost certain (see chapter 16). For instance, say we got 3 789 360 for our last answer. By mistake we put a zero at the end of our answer. The final zero wouldn’t affect our check when casting out nines; we wouldn’t know we had made a mistake.