PRACTICE 9A-9B: Determining & Estimating Irrationals

What is a rational number ?

• A rational number is a number that cannot be written as a fraction.

• A rational number is a number that can be written as a ratio.

• A rational number cannot be a repeating decimal.

Answer explanation

A rational number is a number that can be written as a ratio.

Examples:

$2=\frac{2}{1}$
$0.25\ =\ \frac{1}{4}$
$2.5=\frac{5}{2}$

Which number is RATIONAL?

• $\frac{1}{4}$
  

• $\pi$
  

• $\sqrt{3}$
  

Answer explanation

Rational numbers can be written as a fraction (ratio of two integers).

$\frac{1}{4}$ is exactly that: a ratio of two integers.

_____________ 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:

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

Fill in the blank: _________________ includes integers, fractions, terminating and repeating decimals.

• Rational Numbers

• Irrational Numbers

Answer explanation

Rational numbers (left side of image) include integers, fractions, terminating and repeating decimals.

What type of number is a non-repeating, non-terminating decimal?

• irrational

• prime

• rational

• imaginary

Answer explanation

Irrational numbers cannot be written as a fraction and have a decimal form that never ends, without a repeating pattern.

An irrational number doesn't have a repeating pattern.

• TRUE

• FALSE

Answer explanation

Irrational numbers cannot be written as a fraction and have a decimal form that never ends, without a repeating pattern.

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).

Which number is a perfect square?

• 225

• 132

• 99

• 110

Answer explanation

$\sqrt{225}=15$
225 is the product of a rational number multiplied by itself, so it is a perfect square.

Which number is NOT a perfect square?

• 196

• 169

• 121

• 132

Answer explanation

$\sqrt{196}=14$ perfect root
$\sqrt{169}=13$ perfect root
$\sqrt{121}=11$ perfect root

$\sqrt{132}\approx11.5$ non-perfect root, is irrational

Which number has a square root that is between 7 and 8?

• 48

• 81

• 60

• 36

Answer explanation

$7=\sqrt{49}$
$8=\sqrt{64}$

60 is the only option between these two perfect squares (49 and 64), so $\sqrt{60}$ will be between their perfect roots (7 and 8).