A rational number is a number which can be written as a fraction where both the numerator and denominator are integers. An irrational number is a number which cannot be written as a fraction of two integers.

We can write any terminating or recurring decimal as a fraction, therefore these are rational numbers. However, decimals which are neither terminating nor recurring are irrational numbers.

You might be familiar with one irrational number already: \pi. Like all other irrational numbers, \pi really does go on forever without repeating itself. We say therefore that it doesn't terminate, or repeat.

Another number that is famously irrational is \sqrt{2}. In fact, the square root of most numbers are irrational. If a root is irrational it is called a surd. The square roots of perfect squares are rational, \sqrt{1},\, \sqrt{4},\,\sqrt{9},\,...

A diagram that shows the real number system with different sets of numbers. Ask your teacher for more information.

Examples

Example 1

Is \sqrt{35} rational or irrational?

Worked Solution

Create a strategy

Check whether the number can be written as a fraction.

Apply the idea

35 is not a perfect square, so \sqrt{35} will not equal a whole number.

Using a calculator, we can get\sqrt{35}=5.916079783099616...

The decimals are neither terminating nor recurring. This means that \sqrt{35} cannot be written as a fraction.

So \sqrt{35} is irrational.

Idea summary

A rational number is a number which can be written as a fraction where both the numerator and denominator are integers.

An irrational number is a number which cannot be written as a fraction of two integers.

A surd is a square root which is irrational.

Estimate irrational numbers

As mentioned before the value of \pi, square roots of non-perfect squares and cube roots of non-perfect cubes are examples of irrational numbers. The value of \sqrt 2 is a non-terminating, non-repeating decimal which makes it an irrational number. However, there are ways we can approximate the values without using a calculator.

One way is to consider the nearest integer value as a way to estimate or check our work.

\text{If } a \lt b \text{,}

\text{then } \sqrt{a} \lt \sqrt{b}

Let's say we have a square root, \sqrt{40}. If we ask ourselves what are the closest square numbers that are bigger and smaller than 40, then we'll find that they're 36 and 49. So then we have 36 \lt 40 \lt 49, which leads us to say that \sqrt{36} \lt \sqrt{40} \lt \sqrt{49}. And if we evaluate that further we get 6 \lt \sqrt{40} \lt 7, so we've managed to narrow this square root down to somewhere between 6 and 7.

Examples

Example 2

Represent the following values on the number line:

\pi^2

\sqrt{50}

-1.21

\sqrt{36}

Worked Solution

Create a strategy

Eatimate the values of the numbers using decimals and plot the values on the number line.

Apply the idea

Note that \pi is estimated as 3.14.

\displaystyle \pi^2	\displaystyle \approx	\displaystyle (3.14)^2	Square 3.14
	\displaystyle \approx	\displaystyle 9.9	Evaluate

Find the value of \sqrt{36}.

\displaystyle \sqrt{36}	\displaystyle =	\displaystyle \sqrt{6^2}
	\displaystyle =	\displaystyle 6

Estimate the value of \sqrt {50} by comparing to the closest square numbers that are bigger and smaller than \sqrt {50}.

\sqrt{49}\lt \sqrt{50} \lt \sqrt{64}

7\lt \sqrt{50} \lt 8

The value of \sqrt {50} is closer to \sqrt {49} than \sqrt {64}

Plotting the values on the number line, we have:

Reflect and check

Comparing the numbers, we can see that:

-1.21\lt \sqrt{36} \lt \sqrt{50} \lt \pi^2

Idea summary

We can estimate the values of irrational numbers and represent them on the number line.

To estimate square roots or cubes we can follow these steps:

Determine the closest squares or cubes that are bigger and smaller than the number
Evaluate the square root or cube root to find between what two integers the given number lies.

Estimating square and cube roots:

If a \lt b \text{,}

then, \sqrt{a} \lt \sqrt{b} and

\sqrt[3]{a} \lt \sqrt[3]{b}

Approximate values of irrational numbers

We know that \sqrt{40} is estimated between 6 and 7 because \sqrt{36} <\sqrt{40} < \sqrt{49}.

What if we wanted to approximate \sqrt{40} further? There's a method for that as well. Once you know what integers the square root lies between, you can find the decimal part by using the following formula:

\text{approximation of decimal part } = \dfrac{\text{number inside square root - closest smaller square}}{\text{closest bigger square - closest smaller square}}

Let's use our \sqrt{40} example. The closest smaller square than 40 is 36 while the the closest bigger square is 49. Using the formula:

\displaystyle \text{approximation of decimal part }	\displaystyle =	\displaystyle \dfrac{40 - 36}{49 - 36}	Substite the values in the equation
	\displaystyle =	\displaystyle \dfrac{4}{13}	Evaluate the subtraction
	\displaystyle \approx	\displaystyle 0.3	Evaluate the division

Since \sqrt{40} is estimated between 6 and 7 and the approximation of the decimal part is 0.3, we can say that:

\sqrt{40}\approx 6.3

If you plug this square root into a calculator, you'll see that it is indeed rounded to 6.3. However this method only works well on larger numbers, and the bigger they are the better they'll work. Try and see the difference between using this on say, \sqrt{2} and \sqrt{300}.

We can use the same process for cube roots. Let's take a closer look at the following worked question on how to approximate the decimal part of irrational numbers.

Examples

Example 3

Approximate \sqrt[3]{95} to the nearest tenth without using a calculator.

Worked Solution

Create a strategy

Use the decimal approximation formula to approximate the value.

Apply the idea

First we need to find two perfect cubes close to 95.

\displaystyle 4^3	\displaystyle =	\displaystyle 64	64 \lt 95
\displaystyle 5^3	\displaystyle =	\displaystyle 125	125 \gt 95
\displaystyle 64 \lt 95	\displaystyle <	\displaystyle 125	Compare the values
\displaystyle \sqrt[3]{64} \lt \sqrt[3]{95}	\displaystyle <	\displaystyle \sqrt[3]{125}	Square root numbers
\displaystyle 4 \lt \sqrt{95}	\displaystyle <	\displaystyle 5	Evaluate

Use the decimal approximation formula for the decimal part.

\displaystyle \text{ approximation of decimal part }	\displaystyle =	\displaystyle \dfrac{\text{number inside cube root - closest smaller cube}}{\text{closest bigger cube - closest smaller cube}}
	\displaystyle =	\displaystyle \dfrac{95 -64}{125-64}
	\displaystyle =	\displaystyle \dfrac{31}{61}
	\displaystyle \approx	\displaystyle 0.5
\displaystyle \sqrt[3]{95}	\displaystyle \approx	\displaystyle 4+0.5
	\displaystyle \approx	\displaystyle 4.5

Reflect and check

This method only works well on larger numbers, and bigger they are the better they'll work. There will be a small difference in the actual value. Nevertheless, the method is applicable to estimate the values of irrational numbers.

Idea summary

We can approximate the values of irrational numbers by using the decimal approximation equation for the decimal part, especially of square roots of non perfect squares and cube roots of non perfect cubes.

\text{ approximation of decimal part} = \dfrac{\text{number inside root sign - closest smaller square/cube}}{\text{closest bigger square/cube - closest smaller square/cube}}

Outcomes

AC9M9N01

recognise that the real number system includes the rational numbers and the irrational numbers, and solve problems involving real numbers using digital tools

1.02 Rational and irrational numbers

Ideas

Rational and irrational numbers

Examples

Example 1

Estimate irrational numbers

Examples

Example 2

Approximate values of irrational numbers

Examples

Example 3

Outcomes

AC9M9N01

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