9B: Estimating Irrational Numbers

_____________ number is a number that cannot be written as a ratio of two integers.

• An integer

• A rational

• An irrational

• A repeating

Answer explanation

A rational number cannot be written as a fraction (ratio of two integers). This is because it is non-terminating and non-repeating.

Example:

$\sqrt{2}=1.414213562...$

Between which two whole numbers is the

√20 ? (click the question to better view the image if needed)

• 5 and 6

• 10 and 2

• 4 and 5

• 3 and 4

Answer explanation

20 is between perfect squares 16 and 25.
$16<20<25$

This means the root of 20 is also between the roots of those numbers.
$\sqrt{16}<\sqrt{20}<\sqrt{25}$
$4<\sqrt{20}<5$

The root of 20 is between 4 and 5.

Which point most closely corresponds to the √8 on the number line below? (Click the question to see the picture)

• P

• Q

• R

• S

Answer explanation

8 is between perfect squares 4 and 9.
$4<8<9$
This means the square root of 8 is between the square roots of those numbers.
$\sqrt{4}<\sqrt{8}<\sqrt{9}$
$2<\sqrt{8}<3$
Since 8 is closer to perfect square 9 than it is to perfect square 4, the square root of 8 will also be closer to the square root of 9 (Q in the number line above).

Point B could represent which of the following numbers?

• √24

• √30

• √35

• √42

Answer explanation

B is between whole numbers 5 and 6.
$5<B<6$

B is also much closer to 6 than it is to 5. That means the square of B will also be between the squares of 5 and 6, but closer to the square of 6.
$5^2<B^2<6^2$

$25<B^2<36$

This means $B^2$ is between 25 and 36, but closer to 36.
$\left(B=\sqrt{B^2}\right)$

35 is the closest value to 36 provided without surpassing it.
$B\approx\sqrt{35}$

The √98 falls between what two numbers on a number line?

• 10 and 11

• 11 and 12

• 9 and 10

• 90 and 100

Answer explanation

98 is between perfect squares 81 and 100.
$81<98<100$

This means the square root of 98 is between the square roots of those numbers.
$\sqrt{81}<\sqrt{98}<\sqrt{100}$

$9<\sqrt{98}<10$

List the following numbers from least to greatest:

-64 , 3.5 , √27 , 6.666...

• -64 , 3.5 , √27 , 6.666...

• 3.5 , √27 , 6.666..., -64

• 3.5, -64, √27, 6.666...

• 6.666, √27, -64, 3.5

Answer explanation

-64 is the only negative value, so it is the least.


$\sqrt{25}<\sqrt{27}<\sqrt{36}$

$5<\sqrt{27}<6$
$\sqrt{27}$ is closer to 5 than 6, since 27 is closer to 25 than 36). This is greater than 3.5 and less than 6.666...

Least to greatest:  $-64,\ 3.5,\ \sqrt{27},\ 6.666...$

Put √7 in the right blank so that the numbers are in numerical order.

• __, 2.1, 2.3, 2.9

• 2.1, __, 2.3, 2.9

• 2.1, 2.3, __, 2.9

• 2.1, 2.3, 2.9, __

Answer explanation

 $4<7<9$  
 $\sqrt{4}<\sqrt{7}<\sqrt{9}$  
 $2<\sqrt{7}<3$  
7 is closer to 9 than 4, so the square root of 7 is also closer to the square root of 9 (3). However, it is not SIGNIFICANTLY closer to 9.

2.1, 2.3, √7 , 2.9

What is the approximate value of √12 ?

• 2

• 3.5

• 4.5

• 6

Answer explanation

$9<12<16$
$\sqrt{9}<\sqrt{12}<\sqrt{16}$
$3<\sqrt{12}<4$

3.5 is the only value provided that falls within this range.

What is the approximate value of √28 ?

• 5.6

• 5.9

• 5.3

• 8

Answer explanation

 $25<28<36$     *closer to 25
 $\sqrt{25}<\sqrt{28}<\sqrt{36}$      *closer to  $\sqrt{25}$  
 $5<\sqrt{28}<6$      *closer to 5

 $\sqrt{28}\approx5.3$      *Closest option provided to 5

What is the approximate value of  $\sqrt{37}$  ? (round your approximation to the tenths place)

Answer explanation

 $36<37<49$  
 $\sqrt{36}<\sqrt{37}<\sqrt{49}$  
 $6<\sqrt{37}<7$  

37 is much closer to 36, so  $\sqrt{37}$  will be much closer to 6.

 $\sqrt{37}\approx6.1$