Skill: Approximate irrational numbers

An irrational number cannot be written as a terminating or repeating decimal. To approximate one, find two nearby rational numbers that bound it, then round to the requested place value.

Example: Approximate $\sqrt{2}$ to the nearest hundredth

We know:

$$1^2=1 \quad\text{and}\quad 2^2=4$$

so $\sqrt{2}$ is between $1$ and $2$.

1. Try hundredths:

$$1.41^2=1.9881$$

and

$$1.42^2=2.0164$$

Since $1.9881<2<2.0164$, it follows that

$$1.41<\sqrt{2}<1.42.$$

2. To decide whether to round to $1.41$ or $1.42$, check the thousandths:

$$1.414^2=1.999396$$

and

$$1.415^2=2.002225.$$

Therefore,

$$1.414<\sqrt{2}<1.415.$$

3. The number is less than $1.415$, so it rounds down to the nearest hundredth:

$$\boxed{\sqrt{2}\approx 1.41}$$

The symbol $\approx$ means “approximately equal to.” The decimal $1.41$ is an approximation, not the exact value of the irrational number $\sqrt{2}$.