Study S1 Mathematics Maths - Integers, Rational Numbers, Real Numbers - Geniebook

Integers, Rational Numbers And Real Numbers

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

Rational Numbers

  • 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:

It is a number which consists of a whole number and a proper fraction.

Improper fractions are expressed as one whole and a proper fraction, where one whole is an integer.

 

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 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.

  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.  

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

 

  1.  

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

 

  1.  

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

 

  1.  

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

 

  1. Multiplication and Division of fractions and mixed numbers 

  Proper & Improper fractions Mixed Numbers
Multiplication \(\begin{align*} \frac25 \times \frac34&=\frac{2 × 3}{5 × 4}\\             &=\frac6{20}\\ &=\frac3{10} \end{align*}\) \(\begin{align*} 1\frac25 \times \frac34 &= \frac75 × \frac34 \\ &= \frac{7 × 3}{ 5 × 4} \\ &=\frac{21}{20}\\ &=1\frac1{20} \end{align*}\)
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.

  1. \(\frac23 × (-\frac14)\\\)
  2. \(-\frac25 × (-\frac12)\\\)
  3. \(-3\frac12 × (-\frac13)\\\)

 

Solution: 

  1.  

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

 

  1.  

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

 

  1.  

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

 

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