Fig. 1. Abacus and notations.
SOME OPERATIONS OF ALGEBRA
One of the operations of algebra that is important for a mechanical brain is approximation, the problem of getting close to the right value of a number. Take, for example, finding square root (see the end of this supplement). The ordinary process taught in school is rather troublesome. We can set down another process, however, using a desk calculator to do division, which gives us square root with great speed.
Suppose that we want to find the square root of a number N, and suppose that we have x₀ as a guessed square root correct to one figure. For example, N might be 67.2 and x₀ might be 8, chosen because 8 × 8 is 64, and 9 × 9 is 81, and it seems as if 8 should be near the square root of 67.2. Here is the process:
- 1. Divide x₀ into N, and obtain N / x₀.
- 2. Multiply x₀ + N / x₀ by 0.5 and call the result x₁.
Now repeat:
- 1. Divide x₁ into N and obtain N / x₁.
- 2. Multiply x₁ + N / x₁ by 0.5 and call the result x₂.
Every time this process is repeated, the new x comes a great deal closer to the correct square root. In fact it can be shown that, if x₀ is correct to one figure, then:
| Approximation | Is Correct To ··· Figures |
|---|---|
| x₁ | 2 |
| x₂ | 4 |
| x₃ | 8 |
| x₄ | 16 |
Let us see how this actually works out with 67.2 and a 10-column desk calculator.