Hong Kong
Stage 1 - Stage 3

# Approximate and compare irrational numbers

Lesson

As we have seen before, surds can be manipulated into simpler forms. For example, $\sqrt{250}$250 can be rewritten as $5\sqrt{10}$510.  When we rewrite a surd in this way we do not lose any accuracy, we are simply changing the way we have expressed the surd by finding a factor of $250$250 that is a square number, and then evaluating it. Therefore, we can say that $5\sqrt{10}$510 still has the exact value of $\sqrt{250}$250

However, sometimes we want to give an answer in decimal form to understand how big it is more easily. When we do this, we will need to use a calculator. Try typing $\sqrt{250}$250 or $5\sqrt{10}$510 into a calculator, and you will get $15.8113883$15.8113883. You should note that this is not the exact value! The calculator has given you a rounded number, as it can only display a certain number of digits. In reality, $5\sqrt{10}$510 is an irrational number! We may not need to be this accurate as the calculator though, and we are often asked to round to 2 decimal places. So $\sqrt{250}$250 is almost equivalent to $15.81$15.81. We could write this as $\sqrt{250}$250 ≈ $15.81$15.81

There're also some ways to approximate surds without using a calculator directly. One way is to consider the nearest integer value as a way to check our workings. This is thanks to a rule that states: