Math
submitted by: Jay Jones
George LeBaron – More Than One Way to Multiply 21 x 19
I remember taking a freshman astronomy class from George LeBaron at Southern Utah State College, now Southern Utah University.
One day, George asked the class if they had any questions on the previous homework assignment, and one of the homework problems was brought up. George started working through the problem, which included a calculation for 21 multiplied by 19. George said, “Let’s do that multiplication the old fashioned way, by hand.”
21
x 19
-----
Pointing to the digits as he went along, George said:
“9 times 1” (the one’s digit of 21) “is 9.” And he wrote down a 9 in the one’s column. So far, so good.
21
x 19
------
9
“9 times 1” (the ten’s digit of 19) “is 9.” And he wrote down a 9 in the ten’s column. That was kind of strange.
21
x 19
-----
99
“And 2 add 1” (both ten’s digits) “is 3.” And he wrote down the 3 in the hundred’s column. That was really strange.
21
x 19
-----
399
“The answer is three hundred ninety-nine.”
About one-third of the class was paying no attention at all.
About one-third of the class was smiling, knowing that George was just goofing off.
About one-third of the class was upset. “That’s not the right way to multiply numbers!” they protested.
“What’s the matter, isn’t that the right answer?” asked George.
“Yes, but you did the calculation wrong,” the concerned students said.
“What did I do wrong?” George asked. “I got the right answer, didn’t I?”
The concerned students couldn’t say what was wrong, it just wasn’t the way they had been taught to do multiplication.
The point George was making is that it is important to go through the right process to get the right answer, AND also to know WHY that process is correct.
Another point is that there is more than one way to arrive at the correct answer.
In this case, George knew that the answer to 21 times 19 is 399, and he just went through a goofy way to present to the class that the answer is 399.
But how did he know the answer was 399? Did he memorize that? Probably not. Did he come to class planning this whole presentation? Maybe, but maybe not. Is there another way to arrive at the correct answer that is just as legitimate as the old “by hand” method?
What is a problem that is easy to solve and close to 21 x 19? One Answer: 20 x 20 = 400.
What is the average of 21 and 19? Answer: 20
What is the difference between 21 and 20 and between 20 and 19? Answer: 1
What is (20 x 20) – (1 x 1)? Answer: 399. What is (21 x 19)? Answer: 399.
Can we formalize this into an equation, and will it work for other numbers?
We can find the answer to one number multiplied by another number by squaring the average of the two numbers, minus the square of one-half of the difference:
__ __ __ _ _
| (X+Y) (X + Y) | | (X-Y) (X - Y) |
X * Y = | -------- * -------- | - | ------- * ------- |
| 2 2 | | 2 2 |
-- -- -- --
which reduces to
__ __ __ __
| X² + 2XY + Y² | | X² - 2XY + Y² |
X * Y = | --------------- | - | ---------------- |
| 4 | | 4 |
-- -- -- --
further reducing to
2XY + 2XY
X * Y = ------------- = X * Y
4
One can verify that this works for all real numbers. For an odd number multiplied by an even number, it is quite messy. For numbers with fractions, it is very messy. But for some numbers, it is easier to calculate mentally using this method than to punch the numbers into a calculator.
For example:
22 x 18 = (20 x 20) – (2 x 2) = 396
33 x 27 = (30 x 30) – (3 x 3) = 891
51 x 49 = (50 x 50) – (1 x 1) = 2,499