Question Video: Comparing a Real Number with the Square Root of Its Square Mathematics • Second Year of Preparatory School

Video Transcript

Is the square root of 𝜋 minus 38 squared greater than, equal to, or less than 𝜋 minus 38?

In this question, we are given two real numbers and asked to compare the size of these two real numbers. We might be tempted to instantly answer this question as equal to, since we can say that 𝑥 equals 𝜋 minus 38. We then see that we are comparing the size of the square root of 𝑥 squared with 𝑥. And it may feel like the square root of 𝑥 squared should be 𝑥; however, this is not the case.

To see this, let’s consider the square root of negative one squared. We can evaluate this by first calculating that negative one squared is one, giving us the square root of one. We can then calculate that it is equal to one, which is greater than negative one. The reason for this is that the square root function always takes the nonnegative square root. Hence, we can say that for any real number 𝑥 the square root of 𝑥 squared is equal to the absolute value of 𝑥. This then gives us that the square root of 𝑥 squared is greater than or equal to 𝑥, with equality when 𝑥 is nonnegative.

We can now use these results to compare the sizes of the two expressions in the question. First, we note that 𝜋 minus 38 is negative, since 𝜋 is approximately 3.14. This means that 38 is much larger than 𝜋, so subtracting this value from 𝜋 leaves us with a negative value. We can now use our result of the square root of 𝑥 squared being equal to the absolute value of 𝑥 to evaluate this expression. We substitute 𝑥 equals 𝜋 minus 38 into the formula to obtain that the square root of 𝜋 minus 38 squared is equal to the absolute value of 𝜋 minus 38.

We can now recall that for any nonpositive number 𝑥, the absolute value of 𝑥 is equal to negative 𝑥. So, we can rewrite the absolute value of 𝜋 minus 38 as negative one times 𝜋 minus 38. We can then distribute the negative and simplify to get 38 minus 𝜋. We can then see that this value is greater than 𝜋 minus 38 since it is positive, whereas we have already shown that 𝜋 minus 38 is negative. Hence, the answer is greater than.