A Variation On The Explanation
A nice trick for very young children to learn to add nine is to add ten to the digit and to count back one. Since we are adding 1 to the ten's digit and subtracting one from the unit's digit, the sum of the digits should remain the same. For example 9+2=11 with 1+1=2. When adding 9 to itself, we would thus expect the sum of the digits to be 9 as follows: 9+9=18 (1+8=9) and 9+9+9=27 (2+7=9). Let us look at a simple multiplication: 5×7=35 (3+5=8). Now consider (7+9)×5=16×5=80 (8+0=8) or 7×(9+5)=7×14=98 (9+8=17 1+7=8).
Any positive integer can be written as 9 × n + a where 'a' is a single digit 0 to 8 and 'n' is any positive integer. Thus, using the distributive rule (9 × n + a)×(9 × m + b)= 9 × 9 × n × m + 9 ×(am+bn) +ab. Since the first two factors are multiplied by 9, their sums will end up being 9 or 0, leaving us with 'ab'. In our example, 'a' was 7 and 'b' was 5. We would expect in any base system the number before that base would behave just like the nine.
Read more about this topic: Casting Out Nines
Famous quotes containing the word explanation:
“Natural selection, the blind, unconscious, automatic process which Darwin discovered, and which we now know is the explanation for the existence and apparently purposeful form of all life, has no purpose in mind. It has no mind and no mind’s eye. It does not plan for the future. It has no vision, no foresight, no sight at all. If it can be said to play the role of the watchmaker in nature, it is the blind watchmaker.”
—Richard Dawkins (b. 1941)