Doubling and halving
Knowing addition and subtraction and how to double and halve a multi-digit number is sufficient for doing multiplication and division without learning any multiplication facts. Here we describe algorithms for these two operations that require only arithmetic skills with whole numbers up to nine.
Required skills
1. Knowing the whole part, H(n),
of one half of a number n up to nine:
|
Number n |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
one half of n |
0 |
1/2 |
1 |
1 1/2 |
2 |
2 1/2 |
3 |
3 1/2 |
4 |
4 1/2 |
|
H(n) |
0 |
0 |
1 |
1 |
2 |
2 |
3 |
3 |
4 |
4 |
(H(n) is the whole part of one half of n.)
2.
Knowing the digit of the ones place (the units), D(n),
in the double of a number n.
|
Number n |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
double of n |
0 |
2 |
4 |
6 |
8 |
10 |
12 |
14 |
16 |
18 |
|
D( n) |
0 |
2 |
4 |
6 |
8 |
0 |
2 |
4 |
6 |
8 |
(D(n) is the ones digit of the double of n.)
3. Recognizing even and odd one-digit numbers.
4. Recognizing whether a number is smaller than, or greater than or equal to, 5.
According to an old custom we will write half of the number above the number, and its double below. For example, if 7450789 is given, we have,
half
3725394.5
number n→
7450789.
double
14901578.
Rule for halving
In the number n, look at a digit x and its left neighbor y (blanks are counted as 0), ...yx... .
If y is even, compute z = H(x), and if y is odd, compute z = H(x) + 5.
Write z above x.
z
...yx...
This rule may be used in any order, left to right, right to left or even in random order.
This rule is easy to justify. When y is an odd number it will contribute 5 to the digit z. If it is even it contributes nothing.
Rule for doubling
Look at a digit x and its right neighbor y (blanks are counted as 0), ...xy... .
If y < 5, compute, z = D(x) (the ones digit of the double of x) ,
and if 5 ≤ y, compute z = D(x) + 1, the ones digit of the double of x, plus one).
Write z above x.
z
...xy...
This rule may also be used in any order, left to right, right to left, or even in random order.
This rule is also easy to justify. When y ≥ 5, it will contribute 1 (a carry) to the digit z. If y < 5, it contributes nothing.
Remark
This process is easy to iterate. And if we want to multiply by five instead of dividing by two, we simply move the decimal point one place to the right.
Example: n =
135079.
| 25*n | 3376975. |
| 5*n | 675395. |
| n | 135079. |
| 2*n | 270158. |
| 4*n | 540316. |
| 8*n | 1080632. |
Task.
Practice both algorithms until you can double and halve multi-digit numbers accurately. Do it both with paper and pencil and in a word processor.
At the beginning you may use the tables for H(n) and D(n) as look-up tables.