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:

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

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 

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 

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.

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