How do you work out square root and squares of numbers?

Answer 1

To square a number multiply it by itself (or use a calculator's square button).

For square roots they can be worked out:

1) using a calculator's square root button;

2) take the log of the number (using a calculator or tables), divide it by two and then find the antilog (to the same base, using a calculator or tables) of this result;

3) a kind of long "division":

3.1) write the number in pairs of digits starting either side of the decimal point; if there is an odd number of digits before the decimal point, the first "pair" will be a single digit; if there is an odd number of digits after the decimal point, add a zero to make the last "pair" two digits, eg 123.456 becomes 1 23 . 45 60;

3.2) write in a "bus stop" division over the number (to make it the "dividend"), but extend the down-stroke down a few lines and put a decimal point in the answer over the decimal point in the number;

3.3) find the smallest number that when squared is not greater than the first pair of digits

3.4) write this number to the left of the down-stroke (as the divisor in a division) and over the first pair of digits

3.5) write the square of this number under the first pair of digits and subtract

3.6) bring down the next pair of digits from the "dividend"

3.7) double the answer so far (ignore the decimal point) and write to the left of the down-stroke next to the number formed in step 3.6, leaving room to add a units digit

3.8) find the digit than when added as the units digit to the "divisor" just written (in step 3.7) and the whole "divisor" is multiplied by this digit is not greater than the number formed in step 3.6;

3.9) write this digit as the units digit and over the pair of digits brought down

3.10) multiply the whole divisor by this digit (its units digit) and subtract from the number formed in step 3.6;

3.11) repeat from step 3.6 until there are no more pairs to bring down and the result of the last subtraction is 0, or until the required accuracy is found.

eg √2:

_______1_._4__1__4_2

______-------------------

____1_| 2 . 00 00 00 00

______| 1

______|---

___24_| 1__00

______| ___96

______|--------

__281_|_____4 00

______|_____2 81

______|_____------

_2824_|_____1 19 00

______|_____1 12 96

______|_____----------

28282_|_______6 04 00

______|_______5 65 64

______|_______---------

______|________ 38 36

etc

→ √2 ≈ 1.414