Image via Wikipedia
It is the obligation of every elementary school teacher to motivate and enrich students about the wonders of mathematics. This can be done with some history and some off-the-beaten-path topics as we offer here.
Students will be fascinated to learn that the first occurrence in Western Europe of the Hindu-Arabic numerals we use today was in 1202 in the book "Liber Abaci" by Leonardo of Pisa (otherwise known as Fibonacci). This merchant traveled extensively throughout the Middle East and in the first chapter states that:
These are the nine figures of the Indians: 9, 8, 7, 6, 5, 4, 3, 2, 1. With these nine figures, and with the symbol 0, which in Arabic is called zephirum, any number can be written, as will be demonstrated below.
With this book the use of these numerals was first publicized in Europe. Before that the Roman numerals were used. They were, clearly, much more cumbersome. Take a moment to have students ponder how they would do their calculations if all they had at their disposal were the Roman numerals.
Fibonacci, fascinated by the arithmetic calculations used in the Islamic world, first introduced the system of "casting out nines" as a check for arithmetic in this book. Casting out nines means taking bundles of nine away from the sum, or subtracting a specific number of nines from this sum. Even today it still comes in useful. However, the nice thing about it is that it again demonstrates a hidden magic in ordinary arithmetic.
Before we discuss this arithmetic-checking procedure, we will consider how the remainder of a division by 9 compares to removing nines from the digit sum of the number. Let us find the remainder when 8,768 is divided by 9. The quotient is 974 with a remainder of 2.
This remainder can also be obtained by casting out nines from the digit sum of the number 8,768: 8+7+6+8 = 29, again casting out nines: 2+9 = 11, and again: 1+1 = 2, which was the remainder from before.
Consider the product 734 x 879 = 645,186. We can check this by division, but that would be somewhat lengthy. We can see if this could be correct by casting out nines. Take each factor and the product and add the digits, and then add the digits if the sum is not already a single digit number. Continue this until a single digit number is reached.
For 734: 7+3+4 = 14; then 1+4 = 5For 879: 8+7+9 = 24; then 2+4 = 6For 645,186: 6+4+5+1+8+6 = 30Since 5 x 6 = 30, which yields 3 (casting out nines: 3+0 = 3), is the same as for the product, the answer could be correct.
For practice, have students do another casting out nines "check" for the following multiplication:
56,589 x 983,678 = 55,665,354,342
For 56,589: 5+6+5+8+9 = 33; 3+3 = 6
For 983,678: 9+8+3+6+7+8 = 41; 4+1 = 5
For 55,665,354,342: 5+5+6+6+5+3+5+4+3+4+2 = 48; 4+8 = 12; 1+2 = 3
To check for possibly having the correct product: 6 x 5 = 30 or 3+0 = 3, which matches the 3 resulting from the product digits.
The same scheme can be used to check the likelihood of a correct sum or quotient, simply by taking the sum (or quotient) and casting out nines, taking the sum (or quotient) of these "remainders" and comparing it with the remainder of the sum (or quotient). They should be equal if the answer is to be correct.
The number nine has another unusual feature, which enables us to use a surprising multiplication algorithm. Although it is somewhat complicated, it is nevertheless fascinating to see it work and perhaps try to determine why this happens. This procedure is intended for multiplying a number of two digits or more by 9.
It is best to discuss the procedure with your students in context: Have them consider multiplying 76,354 by 9.
Step 1: Subtract the units digit of the multiplicand from 10.
10 - 4 = 6
Step 2: Subtract each of the remaining digits (beginning with the tens digit) from 9 and add this result to the previous digit in the multiplicand. (For any two digit sums, carry the tens digit to the next sum.)
9 - 5 = 4, 4+4 = 89 - 3 = 6, 6+5 = 11, 19 - 6 = 3, 3+3 = 6, 6+1 = 79 - 7 = 2, 2+6 = 8
Step 3: Subtract 1 from the left-most digit of the multiplicand.
7 - 1 = 6
Step 4: List the results in reverse order to get the desired product.
687,186
Although it is a bit cumbersome, especially when compared to the calculator, this algorithm provides some insights into number theory. But, above all, it's cute!
Leave a comment