# Integers, Rational Numbers And Real Numbers

Integers: These are whole numbers, including positive numbers, negative numbers, and zero. Examples: $-3$$0$$7$.

Rational Numbers: These are numbers that can be expressed as fractions, where the numerator and denominator are both integers. Examples: $\displaystyle\frac{3}{4}$​, $\displaystyle-\frac{2}{5}$, $0.25$ (which is $\displaystyle\frac{1}{4}$​ in fraction form).

Real Numbers: This includes all rational numbers and irrational numbers. Rational numbers can be expressed as fractions or decimals, while irrational numbers cannot be expressed as fractions and have non-repeating, non-terminating decimal expansions. Examples: $\large\pi$ (irrational), $\sqrt{2}$ (irrational), $1.5$ (rational).

In this chapter, we will be discussing the below-mentioned topics in detail

• Proper Fractions, Improper Fractions, Mixed Numbers
• Adding and subtracting fractions and mixed numbers
• Adding and subtracting negative fractions and mixed numbers
• Multiplying and dividing positive fractions and mixed numbers
• Combined operations on positive and negative fractions and mixed numbers

## Rational Numbers

A rational number is a number which can be expressed as a fraction. Some examples of rational numbers are as follows:

Proper Fractions Improper Fractions Mixed Numbers
\begin{align} \frac{3}{4} \end{align} \begin{align} \frac{4}{3} \end{align} \begin{align} \frac{4}{3} = 1\frac{1}{3} \end{align}

### Proper Fractions:

A proper fraction is when the value of the numerator is smaller than that of the denominator.

### Improper Fractions:

An improper fraction is when the value of the denominator is bigger than that of the numerator.

### Mixed Numbers:

Improper fractions are expressed as the sum of an integer and a proper fraction.

Example:

\begin{align*} 1\frac13 &= 1 + \frac13\\ &= \frac33 + \frac13 \\ &= \frac43\end{align*}

### Negative Rational Numbers

Proper And Improper Fractions Mixed Numbers
\begin{align} -\frac{3}{4} \end{align} \begin{align} -1\frac{1}{4} \end{align}
\begin{align} \frac{-3}{4} \end{align} \begin{align} \frac{3}{-4} \end{align} \begin{align} -\frac{5}{4} \end{align}

Both \begin{align} \frac{-3}{4} \end{align} or \begin{align} \frac{3}{-4} \end{align} will simplify to give us  \begin{align} -\frac{3}{4} \end{align}.

This is because a negative integer divided by a positive integer would give a negative value. Likewise, a positive integer divided by a negative integer would also give a negative value.

Similarly, if we rewrite the mixed number \begin{align} -1\frac{1}{4} \end{align} into an improper fraction, it will gives us \begin{align} -\frac{5}{4} \end{align} , in which it can be rewritten as  \begin{align} \frac{-5}{4} \end{align} or \begin{align} \frac{5}{-4} \end{align}.

## 1. Addition and subtraction of positive and negative fractions and mixed numbers

Same Denominator Different Denominator

Step 1:
Join
Step 2:  Expand
Step 3: Simplify
\begin{align*} \frac75+\frac15&=\frac{7+1}5\\\\ &=\frac85\\\\ &=1\frac3 5 \end{align*} \begin{align*} \frac75+\frac12&=\frac{2(7)+5(1)}{10}\\\\ &=\frac{14+5}{10}\\\\ &=\frac{19}{10}\\\\ &=1\frac9{10} \end{align*}
Subtraction

Step 1:
Join
Step 2: Expand
Step 3: Simplify
\begin{align*} \frac75-\frac15&=\frac{7-1}5\\\\ &=\frac65\\\\ &=1\frac1 5 \end{align*} \begin{align*} \frac75-\frac12 &= \frac{2(7)-5(1)}{10}\\\\ &=\frac{14-5}{10}\\\\ &=\frac9 {10} \end{align*}

Let’s understand this with the help of some examples:

Question 1:

Without using a calculator, evaluate the following.

1. \begin{align} \frac{1}{3}+(-\frac{1}{4}) \end{align}
2. \begin{align} -\frac{2}{5}-(-\frac{1}{2})\\ \end{align}
3. \begin{align} -3\frac{1}{2}+(-\frac{1}{3}) \end{align}
4. \begin{align} 1\frac{1}{2}+(-2\frac{1}{3}) \end{align}

Solution:

1.

\begin{align*} & \frac13+(-\frac14)\\ &=\frac13-\frac14\\ &=\frac{4(1)\;-\;3(1)}{12}\\ &=\frac{4 -3}{12}\\ &=\frac1{12} \end{align*}

2.

\begin{align*} & -\frac25-(-\frac12)\\&=-\frac25+\frac12\\ &=-\frac25+\frac12\\ &=\frac{2 (-2) + 5 (1)}{10}\\ &=\frac{-4+5}{10}\\ &=\frac{1}{10} \end{align*}

3.

\begin{align*} & -3\frac12+(-\frac13)\\&=-\frac72-\frac13\\ &=\frac{-7}{2}-\frac13\\ &=\frac{3 (-7)\;-\;2 (1)}6\\ &=\frac{-21\;-\;2}6\\ &=\frac{-23}6\\ &=-3\frac56 \end{align*}

4.

\begin{align*} & 1\frac12+(-2\frac13)\\&=\frac32-\frac73\\ &=\frac{3 (3)\;-\;2 (7)}6\\ &=\frac{9\;-\;14}6\\ &=\frac{-5}6\\ &=-\frac56 \end{align*}

## 2. Multiplication and Division of fractions and mixed numbers

  Multiplication Proper & Improper fractions Mixed Numbers \begin{align*} \frac25 \times \frac{3}{4} &=\frac{2 × 3}{5 × 4}\\\\ &=\frac6{20}\\\\ &=\frac3{10} \end{align*} \begin{align*} 1\frac25 \times \frac34 &= \frac75 × \frac34 \\\\ &= \frac{7 × 3}{ 5 × 4} \\\\ &=\frac{21}{20}\\\\ &=1\frac1{20} \end{align*} \begin{align*} \frac25÷\frac34&=\frac25 × \frac43\\\\ &=\frac8 {15} \end{align*} \begin{align*} 1\frac25÷\frac34&=\frac75 × \frac43 \\\\ &=\frac{28}{15} \\\\ &=1\frac{13}{15} \end{align*}

Let’s understand this with the help of some examples:

Question 2:

Without using a calculator, evaluate the following.

1. $\displaystyle{\frac23 × (-\frac14)}$

2. $\displaystyle{-\frac25 × (-\frac12)\\}$

3. $\displaystyle{-3\frac12 × (-\frac13)}$

Solution:

1.

\begin{align*} & \frac {2}{3} \times (-\frac {1}{4})\\\\ &= \frac{2 \times (-1)}{3 \times 4} \\\\ &=\frac{-2}{12} \\\\ &=\frac {-1}{6} \\\\ &= -\frac{1}{6} \end{align*}

2.

\begin{align*} & -\frac {2}{5} \times (-\frac {1}{2}) \\\\ & = \frac {-2}{5}\times\frac{-1}{2}\\\\ &= \frac{(-2) \times (-1)}{5 \times 2} \\\\ &=\frac{2}{10} \\\\ &=\frac {1}{5} \\\\ \end{align*}

3.

\begin{align*} &-3\frac {1}{2} \times (-\frac {1}{3}) \\\\ &= -\frac{7}{3}\times (-\frac{1}{3})\\\\ &=\frac{(-7) \times (-1)}{2 \times 3} \\\\ &=\frac{7}{6} \\\\ &= 1\frac{1}{6} \end{align*}

Continue Learning
Basic Geometry Linear Equations
Number Patterns Percentage
Prime Numbers Ratio, Rate And Speed
Functions & Linear Graphs 1 Integers, Rational Numbers And Real Numbers
Basic Algebra And Algebraic Manipulation 1 Approximation And Estimation

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