Math Core

An irrational number like $\sqrt{2}$ has a decimal that never ends, so you can never write all of it. But you can still find out about how big it is, place it on a number line and compare it with other numbers. This lesson shows you how, using only perfect squares and some careful multiplication.

Step 1: Trap it between two integers

To estimate $\sqrt{n}$ , find the perfect squares just below and just above $n$ .

For $\sqrt{30}$ : the nearest perfect squares are $25 = 5^2$ and $36 = 6^2$ . Since $25 < 30 < 36$ , taking square roots keeps the order:

$\sqrt{25} < \sqrt{30} < \sqrt{36} \quad\Longrightarrow\quad 5 < \sqrt{30} < 6$

You can also guess where it sits in that gap. $30$ is $5$ steps above $25$ and $6$ steps below $36$ , so $\sqrt{30}$ should be a little less than halfway from $5$ to $6$ .

Step 2: Zoom in to the tenths

To get one decimal place, square some tenths and see where $n$ lands. The number line from $5$ to $6$ splits into tenths:

√30 is between 5.4 and 5.5

Test the tenths near your guess:

$5.4^2 = 29.16 \qquad 5.5^2 = 30.25$

Since $29.16 < 30 < 30.25$ , we know $5.4 < \sqrt{30} < 5.5$ . And $30$ is closer to $30.25$ (off by $0.25$ ) than to $29.16$ (off by $0.84$ ), so to the nearest tenth, $\sqrt{30} \approx 5.5$ . (A calculator gives $5.477\ldots$ )

Approximating a square root

Find consecutive perfect squares around $n$ . Their roots trap $\sqrt{n}$ between two integers.
Square decimals between those integers to trap $\sqrt{n}$ between two tenths.
Repeat with hundredths for even more accuracy. Each step gives one more correct digit.

Worked example: Estimating √2 to the hundredths

Find $\sqrt{2}$ to two decimal places without a calculator.

Integers. $1^2 = 1$ and $2^2 = 4$ , so $1 < \sqrt{2} < 2$ .

Tenths. $1.4^2 = 1.96$ and $1.5^2 = 2.25$ , so $1.4 < \sqrt{2} < 1.5$ .

Hundredths. $1.41^2 = 1.9881$ and $1.42^2 = 2.0164$ , so $1.41 < \sqrt{2} < 1.42$ .

So $\sqrt{2} \approx 1.41$ . Each step zooms in by a factor of $10$ :

√2 lies between 1.41 and 1.42

Common mistake

Don't mix up "between which integers" with "closest to the middle." $\sqrt{50}$ is between $7$ and $8$ , but $50$ is only $1$ more than $49$ , so $\sqrt{50} \approx 7.07$ , very close to $7$ , not $7.5$ . Always check how close $n$ is to each perfect square.

Placing and comparing irrational numbers

Once you have decimal approximations, you can put irrational numbers on a number line together with rational ones. You'll need a few that come up often:

number	approximation
$\sqrt{2}$	$1.41$
$\sqrt{3}$	$1.73$
$\sqrt{5}$	$2.24$
$\pi$	$3.14$

Worked example: Ordering real numbers

Order from least to greatest, and place them on a number line: $\sqrt{5}$ , $\ \pi$ , $\ 2.5$ , $\ \sqrt{8}$ .

$\sqrt{5}$ : $2^2 = 4$ and $2.2^2 = 4.84$ , $2.3^2 = 5.29$ , so $\sqrt{5} \approx 2.2$ .
$\sqrt{8}$ : $2.8^2 = 7.84$ and $2.9^2 = 8.41$ , so $\sqrt{8} \approx 2.8$ .
$\pi \approx 3.14$ and $2.5$ is already a decimal.

From least to greatest: $\sqrt{5}$ , $2.5$ , $\sqrt{8}$ , $\pi$ .

Estimating expressions

Approximations also let you estimate expressions that contain irrational numbers.

Worked example: Estimating an expression

Estimate $\pi^2$ to the nearest whole number. Then estimate $2\sqrt{3}$ to the nearest tenth.

$\pi^2$ . Use $\pi \approx 3.14$ : $3.14^2 = 9.8596$ . So $\pi^2 \approx 10$ .

$2\sqrt{3}$ . Use $\sqrt{3} \approx 1.73$ : $2 \times 1.73 = 3.46$ . So $2\sqrt{3} \approx 3.5$ .

Tip

Your estimate should always pass a squaring check. If you claim $\sqrt{90} \approx 9.5$ , square it: $9.5^2 = 90.25$ . That's very close to $90$ , so the estimate is good.

Practice

Practice 1

Between which two consecutive integers is $\sqrt{45}$ ? Enter the smaller integer first.

Separate answers with commas, e.g. 2, -5

Practice 2

Which integer is closest to $\sqrt{98}$ ?

Practice 3

Estimate $\sqrt{11}$ to the nearest tenth.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

$\sqrt{7}$ lies between which two consecutive tenths? Enter the smaller one first.

Separate answers with commas, e.g. 2, -5

Practice 5

Which list is in order from least to greatest?

Practice 6

Estimate $9\pi$ to the nearest whole number.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

How many whole numbers are between $\sqrt{20}$ and $\sqrt{90}$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.