Ray's New Higher Arithmetic · Art. 105 · Unit 7: Properties of Numbers
28. Casting out nines
Goal: Your student finds the excess of nines in any number and uses it to check addition, subtraction, multiplication and division.
You'll need
- The book
- A slate, small whiteboard or scrap paper
- A notebook and pencil
What's in the book
Some properties of the number nine: how to find the excess of 9's by adding digits, why it works, and how to prove each of the four operations with it, with worked checks (835 × 76 and 8915 ÷ 25).
The lesson, step by step
Warm-up
Ask: What is the remainder when 100 is divided by 9? 1000? 10000? (Always 1.) That is the whole secret.
The idea
Read:
Addition, Subtraction, Multiplication, and Division may be proved by "casting out the 9's."
To cast the 9's out of any number, is to divide the sum of the digits by 9
and keep the remainder. Work the book's example: 768945. Add digits, dropping 9s as you go: the excess is 3.
Why it works
Each 10, 100, 1000 is a multiple of 9 plus 1. So each digit leaves itself as a remainder, and the remainders add up to the digit sum.
Proofs
Read:
The sum of the excess of 9's in the several numbers must equal the excess of 9's in their sum.
For multiplication: multiply the excesses of the factors, cast out the 9s, and compare with the excess of the product. Work the book's check of 835 × 76 = 63460: excesses 7 and 4, 7 × 4 = 28, excess 1; and 63460 has excess 1.
Guided practice
Check the book's division example 8915 ÷ 25 = 356, remainder 15: excess of divisor × excess of quotient + excess of remainder should match the excess of the dividend.
Independent practice
Your student checks five earlier answers from the addition, multiplication and division chapters by casting out nines.
Wrap-up
Ask: Can casting out nines miss a mistake? (Yes: if two digits are swapped, or the error is a multiple of 9. It is a check, not a guarantee.)