## CQSRG Online Educational Material## Finding a square root without a calculatorCQSRG (pronounced CQ Surge) has been researching the earthquake seismicity of Eastern Central Queensland since it began operation in 2002. |

The long division method is useful if you only want your accuracy to a few decimal places. If you want accuracy to several decimal places then Newton’s Method is by far the preferred method.

Both methods will be described.

The particular methodology described here is the one that was taught to me by my Father – back in about 1964. The description will be in the form of worked examples.

First I will demonstrate how to find the square root of a perfect square.

Find the square root of 60025; which we know is going to be 245 – we do, don’t we!

Draw a square root symbol (a radical symbol) with the number whose square root we are seeking underneath the horizontal extension of the symbol. Start at the decimal point and write the digits, in both directions away from the decimal point, in groups of two. In our case, there are no decimal digits, so put in one group of two zeros. Put a decimal point above the horizontal radical line, directly above the decimal point in our number. | . /------------- \/ 6 00 25.00 |

Start with the first group of 1 or 2 digits. Find the largest single-digit, the square of which is less than or equal to that group (6). Write that single digit (in this case 2) above the radical line, and its square (4) under the first group. Draw a line under that square, and subtract it from the first group. | 2 . /------------- \/ 6 00 25.00 00 00 | 4 | - | 2 |

Bring down the next group below the last line drawn. This forms the current remainder. Draw a vertical line to the left of the resulting number. Then double the number above the radical symbol line (highlighted), and write it down with an empty space next to it (where the question mark is placed). | 2 . /------------- \/ 6 00 25.00 00 00 | 4 | ---- | 2 00 4?| |

Next determine (without using a calculator) what single-digit number ? should go in the empty space so that forty-? times ? would be less than or equal to 200.
44 x 4 = 176 45 x 5 = 225, so 4 works.Write 4 on top of the radical line. Calculate 4 x 44 without using a calculator, write that below 200, subtract, and bring down the next pair of digits (in this case the digits 25). |
2 4 . /------------- \/ 6 00 25.00 00 00 | 4 | ---- | 2 00 44| 1 76 | ------- | 24 25 |

Now double the number above the line (24), and write the doubled number (48) with an empty space next to it as indicated. | 2 4 . /------------- \/ 6 00 25.00 00 00 | 4 | ---- | 2 00 44| 1 76 | ------- | 24 25 48?| |

Determine (without using a calculator) what single digit number ? should go in the empty space so that four hundred and eighty-? times ? would be less than or equal to 2425.
484 x 4 = 1936, hmm … to small? 485 x 5 = 2425, so 5 works.Write 5 on top of line. Calculate 5 x 485, write that below 2425, subtract, and see that the difference is zero – we have reached the end of the algorithm. |
2 4 5. /------------- \/ 6 00 25.00 00 00 | 4 | ---- | 2 00 44| 1 76 | ------- | 24 25 485| 24 25 | ----- | 00 00 |

We have determined that the square root of 60025 is 245. Try it and prove it. | 245 x245 ---- 1225 980 +490 ----- 60025 (qed) ----- |

Draw a square root symbol (a radical symbol) with the number 2 underneath the horizontal extension of the symbol. Start at the decimal point and write the digits, in both directions away from the decimal point, in groups of two. In our case, there are no decimal digits, so put in one group of two zeros. Put a decimal point above the horizontal radical line, directly above the decimal point in our number 2.00. | . /------------- \/ 2.00 |

Find the largest single-digit, the square of which is less than or equal to that group (2). Write that single digit (1) above the radical line, and its square (1) under the first group. Draw a line under that square, and subtract it from the first group. | 1. /------------- \/ 2.00 | 1 | - | 1 |

Bring down the next group below the last line drawn. This forms the current remainder. Draw a vertical line to the left of the resulting number. Then double the number above the radical symbol line (highlighted), and write it down with an empty space next to it (where the question mark is placed below). | 1. /------------- \/ 2.00 | 1 | - | 1 00 2?| |

Next determine (without using a calculator) what single-digit number ? should go in the empty space so that twenty-? times ? would be less than or equal to 100.
25 x 5 = 125, too big 24 x 4 = 96, so 4 works.Write 4 on top of line (right of the decimal point). Calculate 4 x 24 without using a calculator, write that below 100, subtract, and bring down the next pair of digits (in this case another set of 00 that we add to the 2.00). |
1. 4 /------------- \/ 2.00 00 | 1 | - | 1 00 24| 96 | ---- | 4 00 |

Now double the number above the radical symbol line ignoring the decimal point, and write it down with an empty space next to it (where the question mark is placed below). | 1. 4 /------------- \/ 2.00 00 | 1 | - | 1 00 24| 96 | ---- | 4 00 28?| |

Determine (without using a calculator) what single digit number ? should go in the empty space so that two hundred and eighty-? times ? would be less than or equal to 400.
282 x 2 = 564, too big 281 x 1 = 281, so 1 works.Write 1 on top of line. Calculate 1 x 281, write that below 400, subtract, and get the difference. |
1. 4 1 /------------- \/ 2.00 00 | 1 | - | 1 00 24| 96 | ---- | 4 00 281| 2 81 | ---- | 1 19 |

Now just keep going doing the same thing, as shown here. | 1. 4 1 4 2 1 /---------------- \/ 2.00 00 00 00 00 | 1 | - | 1 00 24| 96 | ---- | 4 00 281| 2 81 | ---- | 1 19 00 2824| 1 12 96 | ------- | 6 04 00 28282| 5 65 64 | ------- | 38 36 00 282841| 28 28 41 | -------- | 10 07 59 … |

- Choose the number for which you want to determine the square root.
- Make a guess as to what the square root may be.
- Apply the Newton Method to get a better guess (see below).
- Keep applying the Newton Method until you get the required number of decimal places remaining unchanged in two successive guesses.
- The final guess is precise to the required number of decimal places.

Approximation 1 |