Number Sets:
- Natural Numbers: \(1,2,3,4,\dots\)
- Whole Numbers: \(0,1,2,3,4,\dots\)
- Integers: \(\dots,-3,-2,-1,0,1,2,3,\dots\)
- Rational Numbers: numbers of the form \(\tfrac{m}{n}\) where \(m,n\) are integers and \(n\neq0\).
Examples of Rational Numbers: \(-\tfrac{25}{3}, \tfrac{10}{7}, -4, 0, 3, \tfrac{32}{3}, \tfrac{67}{5}\).
There are infinitely many rational numbers between any two rational numbers.
Showing Rational Numbers on a Number Line
Example: Show \(\tfrac{7}{3}, 2, -\tfrac{2}{3}\) on a number line.
- \(2\) is marked directly.
- \(\tfrac{7}{3}=2+\tfrac{1}{3}\). Divide each unit into 3 equal parts. The first division after 2 gives \(\tfrac{7}{3}\).
- \(\tfrac{2}{3}\) is the second mark to the right of 0. Its symmetric point to the left of 0 is \(-\tfrac{2}{3}\).
Practice Set 1.1
1) Show on number line: (1) \(\tfrac{3}{2}, \tfrac{5}{2}, -\tfrac{3}{2}\)
(2) \(\tfrac{7}{5}, -\tfrac{2}{5}, -\tfrac{4}{5}\)
(3) \(-\tfrac{5}{8}, \tfrac{11}{8}\)
(4) \(\tfrac{13}{10}, -\tfrac{17}{10}\)
2) Observe a number line diagram and answer: (1) Which number is at B? (2) Which point shows \(\tfrac{7}{4}\)? (3) Is “D denotes \(\tfrac{5}{2}\)” true?
Comparison of Rational Numbers
Rules: For rational numbers \(\tfrac{a}{b},\tfrac{c}{d}\) with \(b,d>0\):
- If \(ad
- If \(ad=bc\), then equal.
- If \(ad>bc\), then greater.
Examples:
Compare \(\tfrac{5}{4}, \tfrac{2}{3}\).
LCM denominator = 12.
\(\tfrac{5}{4}=\tfrac{15}{12},\;\tfrac{2}{3}=\tfrac{8}{12}\).
\(15>8\Rightarrow \tfrac{5}{4}>\tfrac{2}{3}\).
Compare \(-\tfrac{7}{9}, \tfrac{4}{5}\).
Negative is always less than positive.
So \(-\tfrac{7}{9}<\tfrac{4}{5}\).
Compare \(-\tfrac{7}{3}, -\tfrac{5}{2}\).
Compare \(7/3\) and \(5/2\): \(14/6<15/6\). So \(7/3<5/2\).
Hence \(-7/3>-5/2\).
Compare \(\tfrac{3}{5}, \tfrac{6}{10}\).
Simplify: both equal.
So \(\tfrac{3}{5}=\tfrac{6}{10}\).
Practice Set 1.2
Compare and verify on a number line:
- -7, -2
- 0, \(-\tfrac{9}{5}\)
- \(\tfrac{8}{7}, 0\)
- \(-\tfrac{5}{4}, \tfrac{1}{4}\)
- \(\tfrac{40}{29}, \tfrac{141}{29}\)
- \(-\tfrac{17}{20}, -\tfrac{13}{20}\)
- \(\tfrac{15}{12}, \tfrac{7}{16}\)
- \(-\tfrac{25}{8}, -\tfrac{9}{4}\)
- \(\tfrac{12}{15}, \tfrac{3}{5}\)
- \(-\tfrac{7}{11}, -\tfrac{3}{4}\)
Decimal Representation of Rational Numbers
Divide numerator by denominator.
Types:
- Terminating decimal: e.g. \(\tfrac{7}{4}=1.75\).
- Non-terminating recurring decimal: e.g. \(\tfrac{7}{6}=1.1666...=1.\overline{16}\). \(\tfrac{5}{6}=0.8333...=0.\overline{83}\). \(-\tfrac{5}{3}=-1.666...=-1.\overline{6}\). \(\tfrac{22}{7}=3.142857142857...=3.\overline{142857}\). \(\tfrac{23}{99}=0.2323...=0.\overline{23}\).
Every rational number can be expressed as either terminating or non-terminating recurring decimal.
Practice Set 1.3
Write in decimal form:
- \(\tfrac{9}{37}\)
- \(\tfrac{18}{42}\)
- \(\tfrac{9}{14}\)
- \(-\tfrac{103}{5}\)
- \(-\tfrac{11}{13}\)
Irrational Numbers
Numbers that are not rational (cannot be expressed as \(\tfrac{m}{n}\) with integers). Examples: \(\sqrt{2}, \sqrt{3}, \pi\).
Decimal form: non-terminating and non-recurring.
Showing \(\sqrt{2}\) on a number line
- Draw a unit line OA = 1.
- Erect perpendicular AP = 1 at A.
- Join OP. By Pythagoras, \(OP=\sqrt{2}\).
- Draw arc with center O and radius OP to meet number line at Q. Then OQ=\(\sqrt{2}\).
Similarly we can construct \(\sqrt{3}, \sqrt{5}, \sqrt{7}, \dots\).
Practice Set 1.4
- Complete steps to show \(\sqrt{3}\) using \(\sqrt{2}\).
- Show \(\sqrt{5}\) on number line.
- Show \(\sqrt{7}\) on number line.
Answers
Practice Set 1.1
2.(1) \(-\tfrac{10}{4}\) (2) C (3) True
Practice Set 1.2
- -7 < -2
- 0 > -\(\tfrac{9}{5}\)
- \(\tfrac{8}{7} > 0\)
- \(-\tfrac{5}{4} < \tfrac{1}{4}\)
- \(\tfrac{40}{29} < \tfrac{141}{29}\)
- \(-\tfrac{17}{20} < -\tfrac{13}{20}\)
- \(\tfrac{15}{12} > \tfrac{7}{16}\)
- \(-\tfrac{25}{8} < -\tfrac{9}{4}\)
- \(\tfrac{12}{15} > \tfrac{3}{5}\)
- \(-\tfrac{7}{11} > -\tfrac{3}{4}\)
Practice Set 1.3
(1) 0.243... (2) 0.428571... (3) 0.642857... (4) -20.6 (5) -0.846153...