Calculating Square Roots by Hand

Here is an interesting thing to consider.... The square root of 100 is 10; of 10000 is 100; etc. Hence the square root of a number between 1 and 100 is between 1 and 10; the square root of a number between 100 and 10000 is between 10 and 100; etc. A round square number is a number that ends in 0 and is square. We can use these types of numbers to calculate other square roots of numbers.

Here is a pattern to use,

Number ends in 0 then its square root ends in 0 Iff it is a square number
Number ends in 1 then its square root ends in 1 and 9
Number ends in 2 then there is no square root--- the number is irrational
Number ends in 3 then there is no square root--- the number is irrational
Number ends in 4 then its square root ends in 2,8 Iff it is a square number
Number ends in 5 then its square root ends in 5 iff it is a square number
Number ends in 6 then its square root ends in 4 or 6 iff it is a square number
Number ends in 7 then there is no square root-- the number is irrational
Number ends in 8 then there is no square root--- the number is irrational
Number ends in 9 then its square root ends in 3, or 7 iff it is a square number

Here are the round square numbers......

10 x 10 = 100 therefore its square root is 10
20 x 20 = 400 therefore its square root is 20
30 x 30 = 900 therefore its square root is 30
40 x 40 = 1600 therefore its square root is 40
50 x 50 = 2500 therefore its square root is 50
60 x 60 = 3600 therefore its square root is 60
70 x 70 = 4900 therefore its square root is 70
80 x 80 = 6400 therefore its square root is 80
90 x 90 = 8100 therefore its square root is 90
100 x 100 = 10000 therefore its square root is 100

Therefore if the number is between 81 and 10,000 then the square root will be only a two digit answer. Let us do some problems with this.

Lets do the square root of 2025. It has to be between the square root of 1600 and 2500 because the square root of 1600 is 40 and the square root of 2500 is 50 and the number 2025 is between the two round square numbers of 1600 and 2500. It has to start with a 4 logically and cannot start with a 5 because it cannot be above 50. Since it ends in a 5 the square root has to end in a 5. So the answer is 45.

Lets do the square root of 2187. STOP right there! WE see that it ends in a 7 therefore we cannot write it as ratio of two integers..... as whole numbers are rational.

Lets try to do the square root of 7529. It is between the round square numbers of 6,400 and 8,100 so the square root would have to be between 80 and 90. Since it ends in a 9 the answer would have to be a two digit answer starting with an 8 and ending in either a 3 or 7. If you do the math 83 x 83 or 87 x 87 you will NOT get 7529. We can therefore conclude that the answer is irrational.

This is how we calculate square roots in our heads or on paper. We will next try to take square roots of polynomials in our next lesson--- I found this in a book dated 1906-- very interesting.... until then ta ta!

Julie Code