Part 9

Note that under the cursor line we have the original number, 22·2, on A, and from this the number of digits in the root is determined as before.

The plan of finding the square of a number by ordinary multiplication is often very convenient. The inverse process of finding a square root by trial division is not to be recommended.

To obtain a close value of a root or to verify one found in the usual way, the author has, on occasion, adopted the following plan:—Set 1 (or 10) on B to the number on the A scale (L.H. or R.H. as the case may require), and bring the cursor to the number on D. If the root found is correct, the readings on C under the cursor and on D under the index of C, will be in exact agreement.

If 1 on B is placed to a number n on the L.H. A scale, the student will note that while root n is read on D under 1 on C, the root of 10 n is read on D under 10 on B. Hence, if preferred, the number can be taken always on the first scale of A and the root read under 1 or 10 on B, according to whether there is an odd or even number of digits in the number. Obviously the second root is the first multiplied by √(10).

CUBES AND CUBE ROOTS.

In raising a number to the third power, a combination of the preceding method and ordinary multiplication is employed.

TO FIND THE CUBE OF A NUMBER.—Set the L.H. or R.H. index of C to the number on D, and opposite the number ON THE LEFT-HAND scale of B read the cube on the L.H. or R.H. scale of A.

By this rule four scales are brought into requisition. Of these, the D scale and the L.H. B scale are always employed, and are to be read as of equal denomination. The values assigned to the L.H. and R.H. scales of A will be apparent from the following considerations.

Commencing with the indices of C and D coinciding, and moving the slide to the right, it will be seen that, working in accordance with the above rule, the cubes of numbers from 1 to 2·154 (= ∛(10)) will be found on the first or L.H. scale of A. Moving the slide still farther to the right, we obtain on the R.H. A scale cubes of numbers from 2·154 to 4·641 (or ∛(10) to ∛(100)). Had we a third repetition of the L.H. A scale, the L.H. index of C could be still further traversed to the right, and the cubes of numbers from 4·641 to 10 read off on this prolongation of A. But the same end can be attained by making use of the R.H. index of C, when, traversing the slide to the right as before, the cubes of numbers from 4·641 to 10 on D can be read off on the L.H. A scale over the corresponding numbers on the L.H. B scale. Hence, using the L.H. index of C, the readings on the L.H. A scale may be regarded comparatively as units, those on the R.H. A scale as tens; while for the hundreds we again make use of the L.H. A scale in conjunction with the right-hand index of C.

By keeping these points in view, the number of digits in the cube (N) of a given number (n) are readily deduced. Thus, if the units scale is used, N = 3n − 2; if the tens scale, N = 3n − 1; while if the hundreds scale be used, N = 3n. Placed in the form of rules:—

N = 3n − 2 when the product is read on the L.H. scale of A with the slide to the right (units scale).

N = 3n − 1 when the product is read on the R.H. scale of A; slide to the right (tens scale).

N = 3n when the product is read on the L.H. scale of A with the slide to the left (hundreds scale).

With decimals the same rule applies, but, as before, the number of digits must be read as −1, −2, etc., when one, two, etc., cyphers follow immediately after the decimal point.

EX.—Find the value of 1·4^3.

Placing the L.H. index of C to 1·4 on D, the reading on A opposite 1·4 on the L.H. scale of B is found to be about 2·745 [2·744].

EX.—Find the value of 26·4^3.

Placing the L.H. index of C to 26·4 on D, the reading on A opposite 26·4 on the L.H. scale of B is found to be about 18,400 [18,399·744].

EX.—Find the value of 7·3^3.

In this case it becomes necessary to use the R.H. index of C, which is set to 7·3 on D, when opposite 7·3 on the L.H. scale of B is read 389 [389·017] on A.

EX.—Find the value of 0·073^3.

From the setting as before it is seen that the number of digits in the number must be multiplied by 3. Hence, as there is −1 digit in 0·073, there will be −3 in the cube, which is therefore read 0·000389.

The last two examples serve to illustrate the principle of factorising with powers of 10. Thus

0·073 = 7·3 × 10^{−2}; 0·073^3 = 7·3^3 × (10^{−2})^3 = 389 × 10^{−6} = 0·000389.

Cube Root (Direct Method).—One method of extracting the cube root of a number is by an inversion of the foregoing operation. Using the same scales, the slide is moved either to the right or left until under the given number on A is found a number on the L.H. B scale, identical with the number simultaneously found on D under the right or left index of C. This number is the required cube root.

From what has already been said regarding the combined use of these scales in cubing, it will be evident that in extracting the cube root of a number, it is necessary, in order to decide which scales are to be used, to know the number of figures to be dealt with. We therefore (as in the arithmetical method of extraction) point off the given number into sections of three figures each, commencing at the decimal point, and proceeding to the left for numbers greater than unity, and to the right for numbers less than unity. Then if the first section of figures on the left consists of—

1 figure, the number will evidently require to be taken on what we have called the “units” scale—i.e., on the L.H. scale of A, using the L.H. index of C.

If of 2 figures, the number will be taken on the “tens” scale—i.e., on the R.H. scale of A, using the L.H. index of C.