Square Root Explained
This article explains what the mathematical term square root means. It includes the decimal representation of rational numbers, irrational numbers, and the square roots
of pi and e.
Square Root Explained
The square root of a number is whatever number will, when multiplied by itself, be that number. Examples are 3 is the square root of 9 since 3 times 3 equals 9, 7 is the square root of 49 since 7 times 7 equals 49, and 11 is the square root of 121 since 11 times 11 equals 121. Square roots are symbolized ^1/2, so the previous examples are written 9^1/2=3, 49^1/2=7, and 121^1/2=11.
Notice that there are two square roots of any integer (one positive and one negative). So it follows that, using the first example, the two square roots of 9 are 3 and -3 because 3 times 3 is 9, while at the same time -3 times -3 is 9. It follows because a negative integer times a negative integer is always a positive integer.
The inverse of the square root of an integer is the square of that integer. In other words, the inverse of x^2 is (x)^1/2. The square roots of integers that are not squares uses the same idea as the square root of integers, but we have to use infinite decimals. For example, 2^1/2=1.414213, 3^1/2=1.732050, and 8^1/2=2.828427 . The square roots of these numbers are irrational because they cannot be expressed in the form of an integer.
It has been proven that all irrational numbers have infinite decimals that do not repeat a pattern. This means, using the first example this paragraph, that 1.414213 will not
repeat .414213414213414213 . The square root of a decimal is a little harder. The idea is the samethe square root of a decimal is whatever decimal will, when that decimal is multiplied by itself, be that decimal. Examples are, using the notation above, 0.25^1/2=0.5, 0.49^1/2=.7, and 0.81^1/2=0.9.
When multiplying a square root by a square root, multiply the integers inside the radical sign (square root sign), then find the square root of the resulting integer. For example, (8)^1/2 times (6)^1/2 equals (8x6)^1/2=(48)^1/2. This can be simplified further because the factorization of 48 can be expressed as a product with at least one square: 48=16x3=4^2x3. When this happens the square can be used to simplify the expression. Using the example, (48)^1/2=(16x3)^1/2=(4^2x3)^1/ 2=4(3)^1/2. This is because (4^2)^1/2x(3)^1/2=(4^2x3)^1/2= 2x(3)^1/2.
The square root of a negative number is called an imaginary number. Imaginary numbers are credited to Gauss. The square root of negative one is called i. The notation is i=(-1)^1/2. The square of i is negative one, or ((-1)^1/2)^2=-1. The cube of i is (-1)^1/2 times ((-1)^1/2)^2=(-1)^1/2 times -1=-I (negative i). The fourth power of i is (-1)^1/2 times the cube of I, or (-1)^1/2 times (-i)=(-i^2)=1. The cycle starts over because the fifth power of i is (-1)^1/2 times the fourth power of i, or 1, which should equal i again. The cycle is i, -1, -i, 1, i, -1, -i, 1, .
The number e is 1 plus the reciprocals of the factorials. It can be approximated to seven places using e is approximately equal to 3-(5/63)^1/2=2.7182818 . Also, pi can be approximated to 4 digits using pi is approximately equal to ((40/3)-(12)^1/2)^1/2=3.1415 . The number e to twenty decimal places is 2.71828182845904523536. The square root is approximately 1.648721270700128. The number pi is 3.141592653589793 to fifteen decimal places. The square root of pi is approximately 1.7724538509055.