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Irrational numbers

Adding and subtracting rational numbers

Square numbers and square roots

Here you will learn about rational numbers, including the definition of a rational number, examples of rational numbers and how to identify rational numbers.

Students will first learn about rational numbers as part of the number system in 6th grade.

**Rational numbers** are numbers that can be expressed in the form \cfrac{a}{b} where a and b are **integers** (whole numbers) and b

Rational numbers come in four forms. Below are examples of each. Each example has been expressed as a fraction in the form \frac{a}{b} to show that it is rational.

3.2=\cfrac{16}{5}
Terminating decimals | -7=\cfrac{-7}{1}
Integers |

0 . \overline{3}=\cfrac{1}{3}
Repeating decimals | 4 \cfrac{4}{5} \, =\cfrac{24}{5}
Fractions and mixed numbers |

**A rational number must have a non-zero denominator**

For rational numbers expressed as fractions in the form \cfrac{a}{b}, \; b must be a non-zero integer because zero cannot be a divisor. (Try 5 \div 0 on your calculator and it will give you an error message)

The letter a however can be equal to 0 as you can divide 0 by any real number and get the solution 0. This means that 0 itself is a rational number.

**Numbers that are not rational are called irrational numbers**

If a number cannot be represented as a fraction in the form \cfrac{a}{b} where a and b are integers, then the number is **irrational**.

There are a several famous irrational numbers including

\text { Pi }(\pi=3.141 \ldots) \text {, The Golden Ratio }(\varphi=1.618 \ldots) \text {, and Euler's Number }(e=2.718 \ldots)**Not all fractions are rational numbers**

All rational numbers can be expressed as a fraction, but **not all fractions are rational numbers**.

For example, \cfrac{5}{\sqrt{2}} is a fraction but it is not rational. The numerator is an integer but the denominator is not (the square root of 2 is irrational). Therefore this fraction does not meet the definition of a rational number.

How does this relate to 6th grade math?

**Grade 6 – The Number System (6.NS.C.6)**Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

In order to identify and then show that a number is rational:

**Identify if the number is any of the following. If it is then it is a rational number.**

●**An integer**

●**A terminating decimal**

●**A repeating decimal**

●**A fraction in the form**\bf{\cfrac{\textbf{a}}{\textbf{b}}}**or a mixed number in the form**\bf{C\cfrac{\textbf{a}}{\textbf{b}}}**where**\textbf{a}**and**\textbf{b}**are****integers****Show that the number is rational by writing it as a fraction in the form**\bf{\cfrac{\textbf{a}}{\textbf{b}}}**where**

\textbf{a}**and**\textbf{b}**are integers.**

Assess math progress for the end of grade 4 and grade 5 or prepare for state assessments with these mixed topic, multiple choice questions and extended response questions!

DOWNLOAD FREEAssess math progress for the end of grade 4 and grade 5 or prepare for state assessments with these mixed topic, multiple choice questions and extended response questions!

DOWNLOAD FREEShow that 6.7 is a rational number by expressing it as a fraction in the form \cfrac{a}{b} where a and b are integers.

**Identify if the number is any of the following. If it is then it is a rational number.**

●**An integer**

●**A terminating decimal**

●**A repeating decimal**

●**A fraction in the form**\bf{\cfrac{\textbf{a}}{\textbf{b}}}**or a mixed number in the form**\bf{C\cfrac{\textbf{a}}{\textbf{b}}}**where**\textbf{a}**and**\textbf{b}**are****integers**

6.7 is a terminating decimal and is therefore rational.

2**Show that the number is rational by writing it as a fraction in the form ** \bf{\cfrac{\textbf{a}}{\textbf{b}}} ** where**

\textbf{a} ** and ** \textbf{b} ** are integers.**

6.7 = 6\cfrac{7}{10} = \cfrac{67}{10}

Show that 0.045 is a rational number by expressing it as a fraction in the form \cfrac{a}{b} where a and b are integers.

**Identify if the number is any of the following. If it is then it is a rational number.**

● **An integer**

● **A terminating decimal**

● **A repeating decimal**

● **A fraction in the form** \bf{\cfrac{\textbf{a}}{\textbf{b}}} **or a mixed number in the form ** \bf{C\cfrac{\textbf{a}}{\textbf{b}}} **where** \textbf{a} **and** \textbf{b} **are integers**

0.045 is a terminating decimal and is therefore rational.

**Show that the number is rational by writing it as a fraction in the form ** \bf{\cfrac{\textbf{a}}{\textbf{b}}} ** where ** \textbf{a} ** and ** \textbf{b} ** are integers.**

0.045 = \cfrac{45}{1,000} = \cfrac{9}{200}

Show that -17 is a rational number by expressing it as a fraction in the form \cfrac{a}{b} where a and b are integers.

**Identify if the number is any of the following. If it is then it is a rational number.**

● **An integer**

● **A terminating decimal**

● **A repeating decimal**

● **A fraction in the form** \bf{\cfrac{\textbf{a}}{\textbf{b}}} **or a mixed number in the form ** \bf{C\cfrac{\textbf{a}}{\textbf{b}}} **where** \textbf{a} **and** \textbf{b} **are integers**

-17 is an integer and is therefore rational.

**Show that the number is rational by writing it as a fraction in the form ** \bf{\cfrac{\textbf{a}}{\textbf{b}}} ** where ** \textbf{a} ** and ** \textbf{b} ** are integers.**

-17=\cfrac{-17}{1}

Show that 3\cfrac{4}{5} is a rational number by expressing it as a fraction in the form \cfrac{a}{b} where a and b are integers.

**Identify if the number is any of the following. If it is then it is a rational number.**

● **An integer**

● **A terminating decimal**

● **A repeating decimal**

● **A fraction in the form** \bf{\cfrac{\textbf{a}}{\textbf{b}}} **or a mixed number in the form ** \bf{C\cfrac{\textbf{a}}{\textbf{b}}} **where** \textbf{a} **and** \textbf{b} **are integers**

3\cfrac{4}{5} is a mixed number in the form C\cfrac{a}{b} where a, b and C are integers, and is therefore rational.

**Show that the number is rational by writing it as a fraction in the form ** \bf{\cfrac{\textbf{a}}{\textbf{b}}} ** where ** \textbf{a} ** and ** \textbf{b} ** are integers.**

3\cfrac{4}{5} = \cfrac{3 \; \times \; 5 \; + \; 4}{5} = \cfrac{19}{5}

Show that 0 . \overline{6} is a rational number by expressing it as a fraction in the form \cfrac{a}{b} where a and b are integers.

**Identify if the number is any of the following. If it is then it is a rational number.**

● **An integer**

● **A terminating decimal**

● **A repeating decimal**

● **A fraction in the form** \bf{\cfrac{\textbf{a}}{\textbf{b}}} **or a mixed number in the form ** \bf{C\cfrac{\textbf{a}}{\textbf{b}}} **where** \textbf{a} **and** \textbf{b} **are integers**

0 . \overline{6} is a repeating decimal and is therefore rational.

**Show that the number is rational by writing it as a fraction in the form ** \bf{\cfrac{\textbf{a}}{\textbf{b}}} ** where ** \textbf{a} ** and ** \textbf{b} ** are integers.**

0 . \overline{6} = 2 \times 0 . \overline{3} = 2 \times \cfrac{1}{3} = \cfrac{2}{3}

Show that 0 . \overline{4} is a rational number by expressing it as a fraction in the form \cfrac{a}{b} where a and b are integers.

**Identify if the number is any of the following. If it is then it is a rational number.**

● **An integer**

● **A terminating decimal**

● **A repeating decimal**

● **A fraction in the form** \bf{\cfrac{\textbf{a}}{\textbf{b}}} **or a mixed number in the form ** \bf{C\cfrac{\textbf{a}}{\textbf{b}}} **where** \textbf{a} **and** \textbf{b} **are integers**

0 . \overline{4} is a repeating decimal and is therefore rational.

**Show that the number is rational by writing it as a fraction in the form ** \bf{\cfrac{\textbf{a}}{\textbf{b}}} ** where ** \textbf{a} ** and ** \textbf{b} ** are integers.**

0 . \overline{4}=4 \times 0 . \overline{1}=4 \times \cfrac{1}{9}=\cfrac{4}{9}

- Let students spend time exploring rational numbers using a calculator. This gives them an opportunity to notice patterns and also easily convert between fractions and decimals.

- Have students create a number line of their life with rational numbers – their birth starting at 0. They should plot things that happened before their birth with negative rational numbers and things that happened after their birth with positive rational numbers. Challenge them to represent each rational number on their number line in as many ways as possible.

- Just as developing whole number sense takes time and a variety of experiences in elementary school, so does understanding of rational numbers in middle school. Fit in quick reviews to rational numbers throughout the year or make connections as they appear in other standards to help deepen student understanding over time.

**Thinking zero is not a rational number**

Zero is a rational number because it can be written as \cfrac{0}{n} for any value of n other than 0.

**Thinking that all fractions are rational numbers**

All rational numbers can be written as fractions but not all fractions are rational numbers. If the fraction is in the form \cfrac{a}{b} and a and b are integers and b≠ 0 then the number is rational. However if a or b are not integers then the fraction could represent an irrational number.

E.g.

\cfrac{\sqrt{2}}{3} is a fraction which is irrational.

**Non-terminating decimals and rational numbers**

Some non-terminating decimals are irrational and others are rational.

E.g.

\sqrt{3} = 1.73205… \;*This is a non terminating irrational number.*

\cfrac{7}{9} = 0.777… \;*This is a non terminating rational number.*

- Types of numbers
- Absolute value
- Prime numbers
- Natural numbers
- Prime and composite numbers
- Number sets
- Integers
- Even numbers
- Odd numbers
- Whole numbers

1. Which of the following numbers is **not** rational?

\cfrac{2}{7}

\sqrt{27}

7

0.2 \overline{7}

- \cfrac{2}{7} is a fraction in the form \cfrac{a}{b} where a and b are integers
- 0.2 \overline{7} is a repeating decimal.
- 7 is an integer

All of these are types of rational numbers.

\sqrt{27} = 5.196152… This is non terminating decimal which is not repeating and it cannot be expressed as a fraction in the form \cfrac{a}{b} where a and b are integers. It is therefore irrational.

2. Show that -4 is a rational number by expressing it as a fraction in the form \cfrac{p}{q} where p and q are integers.

\cfrac{-4}{4}

\cfrac{1}{-4}

\cfrac{-4}{1}

\cfrac{4}{-4}

Any integer, like -4, can be shown as a fraction by placing it over 1.

-4=\cfrac{-4}{1}

3. Why is 0 . \overline{7} a rational number?

Because it is a repeating decimal

Because it is an integer

Because it is a terminating decimal

Because it is a mixed number

0 . \overline{7} is a non-terminating decimal and a repeating decimal.

Not all non-terminating decimals are rational, but all repeating decimals are rational because they can be expressed as fractions in the form \cfrac{a}{b} where a and b are integers.

0 . \overline{7}=7 \times \cfrac{1}{9}=\cfrac{7}{9}

4. Show that 3.5 is a rational number by expressing it as a fraction in the form \cfrac{p}{q} where p and q are integers.

\cfrac{3}{5}

\cfrac{5}{3}

\cfrac{7}{2}

\cfrac{2}{7}

3.5 = 3\cfrac{5}{10} = 3\cfrac{1}{2} = \cfrac{3 \; \times \; 2 \; + \; 1}{2} = \cfrac{7}{2}

5. Show that 0 . \overline{8} is a rational number by expressing it as a fraction in the form \cfrac{a}{b} where a and b are integers.

\cfrac{0}{8}

\cfrac{0.\overline{8}}{10}

\cfrac{8}{9}

\cfrac{4}{5}

0.\overline{8} = 8 \times 0.\overline{1} = 8 \times \cfrac{1}{9} = \cfrac{8}{9}

6. Which one of these numbers is a rational number that lies between 6.5 and 6\cfrac{2}{3}?

6.7

6.43

6.67

6.6

A number between 6.5 and 6 \cfrac{2}{3} (6 . \overline{6}=6.6666) would fall here on the number line:

6.67 and 6.7 are rational, but too big.

6.23 is rational, but too small.

6.6 is rational and is in between.

Yes, rational numbers can be shown as terminating or repeating decimals.

There are an infinite number of rational numbers within our number system. In fact, even between any two given numbers there are an infinite number of rational numbers.

Yes, there are irrational numbers, which students learn about later in middle school. In high school students also learn about real numbers, imaginary numbers and complex numbers.

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