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 | |
---|---|---|
Addition 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.
- \(\begin{align} \frac{1}{3}+(-\frac{1}{4}) \end{align}\)
- \(\begin{align} -\frac{2}{5}-(-\frac{1}{2})\\ \end{align} \)
- \(\begin{align} -3\frac{1}{2}+(-\frac{1}{3}) \end{align}\)
- \(\begin{align} 1\frac{1}{2}+(-2\frac{1}{3}) \end{align}\)
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
Proper & Improper fractions | Mixed Numbers | |
Multiplication | \(\begin{align*} \frac25 \times \frac{3}{4} &=\frac{2 × 3}{5 × 4}\\\\ &=\frac6{20}\\\\ &=\frac3{10} \end{align*}\) | \(\begin{align*} 1\frac25 \times \frac34 &= \frac75 × \frac34 \\\\ &= \frac{7 × 3}{ 5 × 4} \\\\ &=\frac{21}{20}\\\\ &=1\frac1{20} \end{align*}\) |
Division | \(\begin{align*} \frac25÷\frac34&=\frac25 × \frac43\\\\ &=\frac8 {15} \end{align*}\) | \(\begin{align*} 1\frac25÷\frac34&=\frac75 × \frac43 \\\\ &=\frac{28}{15} \\\\ &=1\frac{13}{15} \end{align*}\) |
Let’s understand this with the help of some examples:
Question 2:
Without using a calculator, evaluate the following.
- \(\displaystyle{\frac23 × (-\frac14)}\)
- \(\displaystyle{-\frac25 × (-\frac12)\\}\)
- \(\displaystyle{-3\frac12 × (-\frac13)}\)
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 | |
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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 |