# How to take Square Roots by Hand

How to Calculate Square Roots

Logarithms make quick of work of square roots when you don’t have a square root button on your calculator. Just take the log of the number, divide by 2, and find the antilog.

e.g. number: 216.9729

log 2.3364054937 = 2.3364054937

(log 216.9729) / 2 = 1.16820274685

antilog 1.16820274685 = 10 ^ 1.16820274685= 14.73

However, assuming that a calculator is not readily available, there is actually a quite simple if sometimes tedious method of figuring out square roots by hand to any given degree of accuracy. This is called algorism.

1) Group the digits of the number you are trying to find the square root of into pairs starting at the decimal point and moving to the left and right. Add trailing zeroes as necessary to keep the number of digits after the decimal point even and/or to increase the accuracy of the calculated square root

e.g. 216.9729 -> 2 16 . 97 29

2) For simplicity of explanation and diagramming, we’ll use a T-chart. The digits of the calculated square root will be kept in the upper right.

___________

e.g. 2 16. 97 29 | .

|

|

3) Begin by looking at the first pair of digits (in this example it would simply be 1) and finding the greatest number whose square is less than or equal to the value of the pair. Write this number in the upper right and calculate its square in the lower right.

___________

e.g. 2 16. 97 29 | 1 .

| 1 x 1 = 1

|

4) Subtract the resulting product from the first pair of digits of the original number, and then drop the next pair down. (This looks suspiciously like long division because involution i.e. extracting a square root actually IS division, only the divisor changes as you go.)

___________

e.g. 2 16. 97 29 | 1 .

– 1 | 1 x 1 = 1

1 16 |

5) Multiply the number in the top right by two and write it in the lower right, followed by a blank space times a blank space (i.e. _ x _ = )

___________

e.g. 2 16. 97 29 | 1 .

– 1 | 1 x 1 = 1

1 16 | 2_ x _ =

6) Find the greatest number such that when it replaces the blank spaces, the resulting product is less than or equal to the value of the number in the lower left. Write it in the top right. (Since you have now reached the decimal point in the original number, a decimal point must also be placed in the square root.)

___________

e.g. 2 16. 97 29 | 14. .

– 1 | 1 x 1 = 1

1 16 | 24 x 4 = 96

7) Repeat steps 4 through 6 until either you obtain your desired accuracy or until the last value in the lower left is zero. (Ignore the decimal points when multiplying the top right by 2.)

___________

e.g. 2 16. 97 29 | 14. .

– 1 | 1 x 1 = 1

1 16 | 24 x 4 = 96

– 96 |

20 97 | 28_ x _ =

___________

2 16. 97 29 | 14.7 .

– 1 | 1 x 1 = 1

1 16 | 24 x 4 = 96

– 96 |

20 97 | 287 x 7 = 2009

– 20 09 |

88 29 | 294_ x _ =

___________

2 16. 97 29 | 14.73 .

– 1 | 1 x 1 = 1

1 16 | 24 x 4 = 96

– 96 |

20 97 | 287 x 7 = 2009

– 20 09 |

88 29 | 2943 x 3 = 8829

– 88 29 |

0

Voila!