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Wize High School Grade 9 Math Textbook > Rational Numbers

Intro to Rational Numbers

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Intro to Rational Numbers


Numbers are used to represent quantities in everyday life.

Examples
  • The temperature on a hot day is 26 °C26\ \degree\text{C} or 79°F79\degree\text{F}
  • The temperature on a cold day is −20°C-20\degree\text{C} or −4°F-4\degree\text{F}
  • A coffee costs $1.45
  • 25\frac{2}{5} of the class wears glasses

A number line is a very helpful tool for organizing numbers

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Types of Numbers

  • Natural Numbers - You count them, starting at 1
  • Examples: 1, 2, 3,...1,\ 2,\ 3,...
  • Represented by
    N
    for natural numbers
  • Whole Numbers - You count them, starting at 0
  • Examples: 0, 1, 2, 3,...0,\ 1,\ 2,\ 3,...
  • Represented by
    W
    for whole numbers
  • Integers - Positive and negative whole numbers
  • Examples: ..., −3, −2, −1, 0, 1, 2, 3,......,\ -3,\ -2,\ -1,\ 0,\ 1,\ 2,\ 3,...
  • We can put these on a number line
  • Represented by
    I
    for integers
  • Rational Numbers - Any number that can be written as a quotient (division) ab\displaystyle \frac{a}{b} where aa and bb are integers and b≠0b\neq 0
  • Examples: 25, −113, 312, 0.5, −1.23, 3.3‾\displaystyle \frac{2}{5},\ -\frac{11}{3},\ 3\frac{1}{2},\ 0.5,\ -1.23,\ 3.\overline{3}
  • Represented by
    Q
    for quotient



Wize Tip
You might hear your teacher call two numbers opposite, this just means that they are negatives of one another.

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Example: Rewriting Rational Numbers

Rewrite the following rational numbers as quotients of integers.

a) 1.651.65

Wize Tip
Since there are 2 decimal places in this number, we multiply by 100100\frac{100}{100} since there are 2 zeros in 100.

*This is allowed because 100100=1\frac{100}{100}=1, so we're just multiplying by 1.

1.65=1.65×100100=1651001.65=\displaystyle 1.65\times\frac{100}{100}=\frac{165}{100}

You can also simplify this to 165100=3320\displaystyle \frac{\cancel{165}}{\cancel{100}}=\frac{33}{20}

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b) −5-5

−5=−5×11=−51\displaystyle-5=-5\times\frac{1}{1}=-\frac{5}{1}

Wize Tip
Most commonly, you will see −ab\displaystyle -\frac{a}{b} rather than −ab or a−b\displaystyle \frac{-a}{b}\ \text{or}\ \frac{a}{-b}.

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c) −0.302-0.302

Wize Tip
Since there are 3 decimal places in this number, we multiply by 10001000\frac{1000}{1000} since there are 3 zeros in 1000.

*This is allowed because 10001000=1\frac{1000}{1000}=1, so we're just multiplying by 1.

−0.302=−0.302×10001000=−3021000 -0.302=\displaystyle -0.302\times\frac{1000}{1000}=-\frac{302}{1000}\

You can also simplify this to −3021000=−151500\displaystyle -\frac{\cancel{302}}{\cancel{1000}}=-\frac{151}{500}

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d) −0.0013-0.0013

Wize Tip
Since there are 4 decimal places in this number, we multiply by 1000010000\frac{10000}{10000} since there are 4 zeros in 10000.

*This is allowed because 1000010000=1\frac{10000}{10000}=1, so we're just multiplying by 1.

−0.0013=−0.0013×1000010000=−1310000-0.0013=\displaystyle -0.0013\times\frac{10000}{10000}=-\frac{13}{10000}

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e) 00

01 or 02 or 0any non-zero integer\displaystyle \frac{0}{1}\text{ or } \frac{0}{2} \text{ or } \frac{0}{\text{any non-zero integer}}

Practice: Rewriting Rational Numbers

Rewrite the following rational numbers as quotients of integers.

a) 1.251.25

b) −0.160-0.160

c) −0.0005-0.0005

Enter your final answer as an improper fraction, make sure to simplify your answer.
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Example: Comparing Rational Numbers

List the following rational numbers from smallest to largest.
a) −3.45, 3.45, −3.451, −3.448, 3.451, 3.448-3.45,\ 3.45,\ -3.451,\ -3.448,\ 3.451,\ 3.448

Wize Tip
When a number is negative, the larger the value looks, the smaller the actual number is.

Example: 5>35>3 but −5<−3-5<-3

Some of the numbers have 2 decimal places, some have 3. Rewrite them so that they have all 3 decimal places:

−3.450, 3.450, −3.451, −3.448, 3.451, 3.448-3.450,\ 3.450,\ -3.451,\ -3.448,\ 3.451,\ 3.448

Now it's much easier to compare them:

−3.451<−3.45<−3.448<3.448<3.45<3.451\boxed{-3.451<-3.45<-3.448<3.448<3.45<3.451}

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b)  13, 23, −45, −23, −13, 45\displaystyle\ \frac{1}{3},\ \frac{2}{3},\ -\frac{4}{5},\ -\frac{2}{3},\ -\frac{1}{3},\ \frac{4}{5}

Wize Tip
When comparing fractions, it helps to convert them to a common denominator.

You can also draw pictures, use a number line, or use fraction tiles to help you see how big each fraction is.

If you're allowed a calculator, you can convert the fractions to decimals first.

A common denominator is 3×5=153\times 5=15:

 1×53×5, 2×53×5, −4×35×3, −2×53×5, −1×53×5, 4×35×3\displaystyle\ \frac{1\colorFour{\times 5}}{3\colorFour{\times 5}},\ \frac{2\colorFour{\times 5}}{3\colorFour{\times 5}},\ -\frac{4\colorFour{\times 3}}{5\colorFour{\times 3}},\ -\frac{2\colorFour{\times 5}}{3\colorFour{\times 5}},\ -\frac{1\colorFour{\times 5}}{3\colorFour{\times 5}},\ \frac{4\colorFour{\times 3}}{5\colorFour{\times 3}}

Now all of the fractions have the same denominator:

 515, 1015, −1215, −1015, −515, 1215\displaystyle\ \frac{5}{15},\ \frac{10}{15},\ -\frac{12}{15},\ -\frac{10}{15},\ -\frac{5}{15},\ \frac{12}{15}

We just need to compare the numerators:

−1215<−1015<−515<515<1015<1215\displaystyle -\frac{12}{15}<-\frac{10}{15}<-\frac{5}{15}<\frac{5}{15}<\frac{10}{15}<\frac{12}{15}

Rewriting this back into our original fractions:

−45<−23<−13<13<23<45\boxed{\displaystyle -\frac{4}{5}<-\frac{2}{3}<-\frac{1}{3}<\frac{1}{3}<\frac{2}{3}<\frac{4}{5}}

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c) −356, −334, −423\displaystyle -3 \frac{5}{6},\ \displaystyle -3 \frac{3}{4},\ -4\frac{2}{3}

Wize Tip
When comparing fractions, it helps to convert them to a common denominator.

You can also draw pictures, user a number line or use fraction tiles to help you see how big each fraction is.

If you're allowed a calculator, you can convert the fractions to decimals first.

A common denominator is 6×4×3=726\times 4\times 3=72:

−35×126×12, −33×184×18, −42×243×24\displaystyle -3 \frac{5\colorFour{\times12}}{6\colorFour{\times12}},\ \displaystyle -3 \frac{3\colorFour{\times18}}{4\colorFour{\times18}},\ -4\frac{2\colorFour{\times24}}{3\colorFour{\times24}}

Now all the fractions have the same denominator:

−36072, −35472, −44872\displaystyle -3 \frac{60}{72},\ -3 \frac{54}{72},\ -4 \frac{48}{72}

When comparing these, we first look at the whole number part, and only use the fraction part if there's a tie:

−44872<−36072<−35472\displaystyle -4\frac{48}{72}<-3\frac{60}{72}<-3\frac{54}{72}

Rewriting this back into our original fractions:

 −423<−356<−334\boxed{\displaystyle\ -4 \frac{2}{3}<-3 \frac{5}{6}<-3 \frac{3}{4}}

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d) 0.3, 15, 0.33, −32, 1.2, −86, 0.329, −1.51\displaystyle 0.3,\ \frac{1}{5},\ 0.33,\ -\frac{3}{2},\ 1.2,\ -\frac{8}{6},\ 0.329,\ -1.51

Wize Tip
To compare rational numbers in different forms, it's easier if we convert them all to fractions or all to decimals.

Let's convert these to decimals:
  • 15=0.2\displaystyle \frac{1}{5}=0.2
  • −32=−1.5\displaystyle -\frac{3}{2}=-1.5
  • −86=−1.33333......repeating\displaystyle -\frac{8}{6}=-1.33333...... \text{repeating}
Now we can more easily compare these rational numbers

−1.51< −32<−86<15<0.3<0.329<0.33<1.2\boxed{\displaystyle-1.51<\ -\frac{3}{2}<-\frac{8}{6}<\frac{1}{5}<0.3<0.329<0.33<1.2}

Practice: Comparing Rational Numbers

In each of the following questions, you are asked to pick the smallest or biggest number in the group.
Pick the smallest number.
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Example: Listing Rational Numbers

Name three decimals and three fractions between the following pairs of rational numbers.

a) 137\displaystyle1\frac{3}{7} and 147\displaystyle1\frac{4}{7}.
Decimal Method
Let's first convert these to decimals:
  • 137=1.42...\displaystyle1\frac{3}{7}=1.42...
  • 137=1.57...\displaystyle1\frac{3}{7}=1.57...
Therefore, three possibles decimals between these two rational numbers are 1.51, 1.52, 1.531.51,\ 1.52,\ 1.53 (there are many other possible answers).

Fraction Method
We can multiply the numerator and denominator by the same integer:
  • 13×107×10=13070\displaystyle1\frac{3\orange{\times10}}{7\orange{\times10}}=1\frac{30}{70}
  • 14×107×10=14070\displaystyle1\frac{4\orange{\times10}}{7\orange{\times10}}=1\frac{40}{70}
Therefore, three possible fractions between these two rational numbers are 13170, 13270, 13370\displaystyle 1\frac{31}{70},\ 1\frac{32}{70},\ 1\frac{33}{70} (there are many other possible ansers).

*Note: you could have chosen to multiply the numerator and denominator by any other integer.

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b) −215\displaystyle -2\frac{1}{5} and −225-\displaystyle 2\frac{2}{5}.
Decimal method
Let's first convert these to decimals:
  • −215=−2.2\displaystyle-2\frac{1}{5}=-2.2
  • −225=−2.4\displaystyle-2\frac{2}{5}=-2.4
Therefore, three possibles decimals between these two rational numbers are −2.21, −2.22, −2.23-2.21,\ -2.22,\ -2.23 (there are many other possible answers).

Fraction method
We can multiply the numerator and denominator by the same integer:
  • −21×105×10=−21050\displaystyle-2\frac{1\orange{\times10}}{5\orange{\times10}}=-2\frac{10}{50}
  • −22×105×10=−22050\displaystyle-2\frac{2\orange{\times10}}{5\orange{\times10}}=-2\frac{20}{50}
Therefore, three possible fractions between these two rational numbers are −21150, −21250, −21350\displaystyle -2\frac{11}{50},\ -2\frac{12}{50},\ -2\frac{13}{50} (there are many other possible answers).

*Note: you could have chosen to multiply the numerator and denominator by any other integer.

Practice: Possible Rational Numbers

The thickness of a wooden plank needed to build a shelf must be between 38"\displaystyle\frac{3}{8}^" and 58"\displaystyle\frac{5}{8}^". At a local lumber store, you're able to find wood with the following thicknesses. Select all of the ones that you can use to build this shelf.