Why does every positive number have two square roots?

Every positive number has two square roots. One square root is positive, while the other is exactly same, but negative. For example, the square root of 100 is +10 and -10.

Why?

To understand this, we need to re iterate the definition of a square root:
"The square root of a number is another number, which on multiplying with itself, will give the original number."
We know that 10 x 10 = 100, so 10 is a square root of 100.

But -10 x -10 is also equal to 100, because on multiplying two negative numbers the result is always a positive number.

Thus both, positive 10 and negative 10, are square roots of the positive number 100.

Principle square root

The positive square root of a number is called its principle square root. For example, 10 and -10 are the two square roots of 100, but 10 is called the principle square root.

Similarly,
  • the positive number 64 has two square roots, negative eight (-8) and positive eight (+8).
  • 36 has two square roots, negative six (-6) and positive six (+6)
  • 49 has two square roots, negative seven (-7) and positive seven (+7).
Generally the positive sign on the left of a number is not written: it is assumed that a number is positive if it does not have a negative sign on its left, so each of the positive square roots written above may be written without the positive signs as well.

We can conclude that,

There are two square roots of a positive number - one square root is positive, whereas one square root is negative.

The square root of a number is generally represented by a "+/-"  symbol before it as follows:
Square root of 100 = +/-100
(where sqrt means symbol of square root)

Now you must be wondering about the square root of negative numbers? Do negative numbers have two square roots as well? Or something different?

Negative numbers are not taken into account in this post because they have no real square roots. In simpler words, you can not calculate the square root of a negative number. This is, because there can not be any positive or negative number, which on being multiplied with itself, will give a negative result, because positive times positive equals positive and negative times negative equals positive as well.

So the square roots of negative numbers are called 'imaginary numbers' in math. These are those numbers that do not come under the set of negative numbers, positive numbers and zero.

Example: Give both the positive and negative square roots of each number:
  • 25
    • Positive square root = 5
    • Negative square root = -5
  • 36
    • Positive square root = 6
    • Negative square root = -6
  • 49
    • Positive square root = 7
    • Negative square root = -7
  • 64
    • Positive square root = 8
    • Negative square root =-8
  • 81
    • Positive square root = 9
    • Negative square root = -9
  • 100
    • Positive square root = 10
    • Negative square root = -10

6 comments:

  1. how about questions below?
    1) the square roots of 25 are:
    2) the square roots of 5^2 are:
    3) the square roots of (-5)^2 are:
    4) sqt 5^2 is 5 and -5, is it correct?

    ReplyDelete
  2. Replies
    1. The positive square root is called the principle one. Square roots of 16 are 4 and -4, of which 4 is called the principle square root. This can not be applied to square roots of negative numbers.

      Thanks for the thoughtful question!

      Delete
  3. Why do numbers have both a positive and a negative square root

    ReplyDelete
    Replies
    1. That is what the post is about. Maybe reading it again (if not already read) will help you answer that question :)

      Delete