Exercise 1.1 Practice
Introduction to Number Systems
Q1: Rational Numbers
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Is the number 0 a rational number? Can you write it in the form $p/q$, where $p$ and $q$ are integers and $q \neq 0$?
Yes, 0 is a rational number.
It can be written in the form $p/q$ where $p = 0$ and $q$ is any non-zero integer.
Examples: $\frac{0}{1}, \frac{0}{2}, \frac{0}{-5}$, etc.
Yes, it can be written as 0/1, 0/2, etc.
Q2: Finding Rational Numbers
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Find five rational numbers between 1 and 2.
Method: To find 5 numbers, we can multiply and divide the integers by $5+1=6$.
$1 = \frac{1 \times 6}{6} = \frac{6}{6}$
$2 = \frac{2 \times 6}{6} = \frac{12}{6}$
The numbers between $\frac{6}{6}$ and $\frac{12}{6}$ are:
$\frac{7}{6}, \frac{8}{6}, \frac{9}{6}, \frac{10}{6}, \frac{11}{6}$.
7/6, 8/6, 9/6, 10/6, 11/6
Q3: Rational Numbers (Fractions)
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Find five rational numbers between $\frac{2}{5}$ and $\frac{3}{5}$.
Method: To find 5 numbers, multiply numerator and denominator by $5+1=6$.
$\frac{2}{5} = \frac{2 \times 6}{5 \times 6} = \frac{12}{30}$
$\frac{3}{5} = \frac{3 \times 6}{5 \times 6} = \frac{18}{30}$
The numbers between $\frac{12}{30}$ and $\frac{18}{30}$ are:
$\frac{13}{30}, \frac{14}{30}, \frac{15}{30}, \frac{16}{30}, \frac{17}{30}$.
13/30, 14/30, 15/30, 16/30, 17/30
Q4: True or False
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State whether the following statements are true or false. Give reasons for your answers.
(i) Every whole number is a natural number.
(ii) Every integer is a rational number.
(iii) Every rational number is an integer.
(i) Every whole number is a natural number.
(ii) Every integer is a rational number.
(iii) Every rational number is an integer.
(i) False: Because 0 is a whole number but not a natural number.
(ii) True: Because every integer $m$ can be expressed in the form $m/1$, and so it is a rational number.
(iii) False: Because $\frac{3}{5}$ is a rational number but not an integer.
(i) False, (ii) True, (iii) False