Exercise 3.4
1. Give two examples of irrational numbers.
Two examples of irrational numbers are:
- [tex]\( \sqrt{2} \approx 1.4142135623730951 \)[/tex]
- [tex]\( \pi \approx 3.141592653589793 \)[/tex]
Irrational numbers are numbers that cannot be expressed as a fraction [tex]\( \dfrac{a}{b} \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are integers and [tex]\( b \neq 0 \)[/tex]. These numbers have non-repeating, non-terminating decimal expansions.
2. Express the following decimal numbers as rational numbers.
(a) 0.24
To express 0.24 as a rational number, we can write it as a fraction. Since 0.24 has two digits after the decimal point, we can write it as:
[tex]\[
0.24 = \dfrac{24}{100}
\][/tex]
We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4:
[tex]\[
\dfrac{24}{100} = \dfrac{24 \div 4}{100 \div 4} = \dfrac{6}{25}
\][/tex]
Therefore, 0.24 expressed as a rational number is [tex]\( \dfrac{6}{25} \)[/tex].
(b) 0.106
To express 0.106 as a rational number, we convert it into a fraction. Since 0.106 has three digits after the decimal point, we can write it as:
[tex]\[
0.106 = \dfrac{106}{1000}
\][/tex]
We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
[tex]\[
\dfrac{106}{1000} = \dfrac{106 \div 2}{1000 \div 2} = \dfrac{53}{500}
\][/tex]
Therefore, 0.106 expressed as a rational number is [tex]\( \dfrac{53}{500} \)[/tex].
