How do you find the square root of 17?

Answer 1

There may be no square root of 17 in integers, but in real numbers (which include decimals) there is. (As √17 is irrational, only an approximation can be made in a finite number of decimals.)

To calculate it, a "long division" method can be used:

1. Write the number in pairs of digits starting with the tens and units; the last "pair" on the left may have only 1 digit (eg 12345 would be written as 1 23 45)
2. If there is a decimal part, write this again in pairs, starting with the tenths and hundredths; If the last "pair" on the right has only 1 digit add a zero (eg 123.456 would be written as 1 23 . 45 60)
3. Extra zeros can be added in pairs at the end of the number after the decimal point (eg 123 would become 1 23 . 00 00 00; 123.456 would become 1 23 . 45 60 00 00 00)
4. Now do the "division" to find the square root:
   1. Find the largest square not greater than the first pair of digits (which may contain only 1 digit)
   2. Write its square root over the digit(s) and subtract the square from the digit(s)
   3. Bring down the next pair of digits (like normal long division, but both digits are appended to the end of the subtraction result)
   4. Double the quotient and multiply it by 10
   5. Now find a digit to add to this (ie become the units digit) so that when the number is multiplied by this digit is it the largest number not bigger than the number formed by step 3
   6. Subtract this product from the number (formed by step 3)
   7. Repeat from step 3 until the result of the subtraction is zero AND there are no more pairs of digits to bring down; OR the required accuracy has been reached.

This is like normal long division except:

• a pair of digits is brought down each time;
• the divisor changes each time.

Now to do it for 17:

................4 . ..1....2...3...1

............------------------- .........Calculating 4 dp to round to 3 dp

.........4 | 17 . 00 00 00 00 .........4^2 is the highest square not bigger than 17

..............16 ...............................= 4 x 4

..............---

.......81 |....1..00 ........................Bring down 00, double 4 = 8 & put 1 in result & on end to make divisor 81

......................81 .......................= 81 x 1

................------- .......................Subtract to give 19

......822 |.......19 00 ...................Bring down 00, double 41 = 82 & put 2 in result & on end to make divisor 822

......................16 44 ...................= 822 x 2

......................------ ...................Subtract to give 256

....8243 |..........2 56 00 ..............Bring down 00, double 412 = 824 & put 3 in result & on end to make divisor 8243

........................2 47 29 ..............= 8243 x 3

........................-------- ..............Subtract to give 871

..82461 |...............8 71 00 .........Bring down 00, double 4123 = 8246 & put 1 in result & on end to make divisor 82461

.............................8 24 61 ..........= 82461 x 1

............................--------- .........Subtract

................................46 39..........Calculated to 4 dp to round to 3 dp.

Thus √17 ≈ 4.123 to 3 dp

(Doubling and putting a digit on the end is a [bad] abbreviated way of saying step 4.4 in the description above: 4 x 2 = 8, 8 x 10 = 80, 80 + 1 = 81; 4 x 2 = 8, 1 on the end makes it 81.)