Rational and Irrational Numbers

Q1: Write the additive inverse of:

i. 5

Step 1: The additive inverse of a number \(a\) is \(-a\), such that \(a + (-a) = 0\).
Step 2: Change the sign of 5 to -5.
Answer: -5

ii. -7

Step 1: To find the additive inverse, change the sign from negative to positive.
Step 2: \(-(-7) = 7\).
Answer: 7

iii. \(\frac{5}{9}\)

Step 1: Change the sign of the fraction.
Step 2: \(-(\frac{5}{9}) = -\frac{5}{9}\).
Answer: -\(\frac{5}{9}\)

iv. \(\frac{-3}{17}\)

Step 1: Change the negative sign to positive.
Step 2: \(-(\frac{-3}{17}) = \frac{3}{17}\).
Answer: \(\frac{3}{17}\)

v. 0

Step 1: Zero is neither positive nor negative.
Step 2: \(0 + 0 = 0\), so the additive inverse of 0 is itself.
Answer: 0

vi. \(11\frac{5}{17}\)

Step 1: This is a positive mixed fraction.
Step 2: Apply a negative sign to the entire value.
Answer: -11\(\frac{5}{17}\)

vii. \(-5\frac{3}{8}\)

Step 1: This is a negative mixed fraction.
Step 2: Change the sign to positive.
Answer: 5\(\frac{3}{8}\)

viii. -37

Step 1: The additive inverse of -37 is 37.
Answer: 37

ix. 1

Step 1: The additive inverse of 1 is -1.
Answer: -1

Q2: Write the multiplicative inverse of:

i. 9

Step 1: The multiplicative inverse (reciprocal) of a number \(a\) is \(\frac{1}{a}\), such that \(a \times \frac{1}{a} = 1\).
Step 2: The reciprocal of 9 is \(\frac{1}{9}\).
Answer: \(\frac{1}{9}\)

ii. -1

Step 1: The reciprocal of -1 is \(\frac{1}{-1}\).
Step 2: \(\frac{1}{-1} = -1\).
Answer: -1

iii. \(\frac{11}{16}\)

Step 1: To find the reciprocal of a fraction, swap the numerator and the denominator.
Step 2: Reciprocal of \(\frac{11}{16}\) is \(\frac{16}{11}\).
Answer: \(\frac{16}{11}\)

iv. \(5\frac{1}{4}\)

Step 1: Convert the mixed fraction to an improper fraction: \(5\frac{1}{4} = \frac{(5 \times 4) + 1}{4} = \frac{21}{4}\).
Step 2: The reciprocal of \(\frac{21}{4}\) is \(\frac{4}{21}\).
Answer: \(\frac{4}{21}\)

v. \(\frac{-2}{3}\)

Step 1: Swap the numerator and denominator: \(\frac{3}{-2}\).
Step 2: Standard form is \(-\frac{3}{2}\).
Answer: \(-\frac{3}{2}\)

vi. \(17\frac{3}{20}\)

Step 1: Convert to improper fraction: \(17\frac{3}{20} = \frac{(17 \times 20) + 3}{20} = \frac{343}{20}\).
Step 2: The reciprocal of \(\frac{343}{20}\) is \(\frac{20}{343}\).
Answer: \(\frac{20}{343}\)

vii. \(-18\frac{1}{2}\)

Step 1: Convert to improper fraction: \(-18\frac{1}{2} = -\frac{(18 \times 2) + 1}{2} = -\frac{37}{2}\).
Step 2: The reciprocal is \(-\frac{2}{37}\).
Answer: \(-\frac{2}{37}\)

viii. -5

Step 1: Write -5 as \(\frac{-5}{1}\).
Step 2: The reciprocal is \(\frac{1}{-5}\) or \(-\frac{1}{5}\).
Answer: \(-\frac{1}{5}\)

ix. \(\frac{-20}{41}\)

Step 1: Swap numerator and denominator: \(\frac{41}{-20}\).
Step 2: Standard form is \(-\frac{41}{20}\).
Answer: \(-\frac{41}{20}\)

Q3: Represent each of the following on the number line:

i. \(\frac{3}{7}\)

Step 1: Since \(\frac{3}{7}\) is between 0 and 1, divide the segment from 0 to 1 into 7 equal parts.
Step 2: Starting from 0, count 3 parts to the right.

            P
←--|--|--|--Φ--|--|--|--|--|--|--→
   0                    1

Answer: Point P represents \(\frac{3}{7}\)

ii. \(\frac{16}{5}\)

Step 1: Convert to mixed fraction: \(\frac{16}{5} = 3\frac{1}{5}\).
Step 2: This point lies between 3 and 4. Divide the segment from 3 to 4 into 5 equal parts.
Step 3: Take the 1st part after 3.

                                  Q
←-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-Φ-|-|-|-|-|-→
  0         1         2         3         4

Answer: Point Q represents \(\frac{16}{5}\)

iii. \(-\frac{4}{9}\)

Step 1: This point lies between 0 and -1. Divide the segment into 9 equal parts.
Step 2: Move 4 parts to the left from 0.

                R
←-|-|-|-|-|-|-|-Φ-|-|-|-|-|-|-→
     -1                 0

Answer: Point R represents \(-\frac{4}{9}\)

iv. \(-\frac{18}{11}\)

Step 1: Convert to mixed fraction: \(-\frac{18}{11} = -1\frac{7}{11}\).
Step 2: This point lies between -1 and -2. Divide the segment into 11 equal parts.
Step 3: Move 7 parts to the left starting from -1.

            S
←-|-|-|-|-|-Φ-|-|-|-|-|-|-|-|-|-→
   -2                    -1

Answer: Point S represents \(-\frac{18}{11}\)

v. \(-3\frac{1}{6}\)

Step 1: This point lies between -3 and -4. Divide the segment into 6 equal parts.
Step 2: Move 1 part to the left from -3.

              T
←-|-|-|-|-|-|-Φ-|-|-|-|-|-|-|-|-|-→
   -4          -3          -2

Answer: Point T represents \(-3\frac{1}{6}\)

Q4: Find a rational number between \(\frac{3}{5}\) and \(\frac{7}{9}\).

Step 1: Use the formula for the rational number between \(a\) and \(b\): \(\frac{1}{2}(a + b)\).
Step 2: Add the two fractions: \(\frac{3}{5} + \frac{7}{9}\).
L.C.M. of 5 and 9 is 45.
\(\frac{3 \times 9}{45} + \frac{7 \times 5}{45} = \frac{27 + 35}{45} = \frac{62}{45}\).
Step 3: Multiply the sum by \(\frac{1}{2}\):
\(\frac{1}{2} \times \frac{62}{45} = \frac{31}{45}\).
Answer: \(\frac{31}{45}\)

Q5: Find two rational numbers between:

i. 2 and 3

Step 1: To find two numbers, we can use the denominator \(n+1 = 2+1 = 3\). Convert 2 and 3 into fractions with denominator 3.
\(2 = \frac{2 \times 3}{3} = \frac{6}{3}\)
\(3 = \frac{3 \times 3}{3} = \frac{9}{3}\)
Step 2: Numbers between \(\frac{6}{3}\) and \(\frac{9}{3}\) are \(\frac{7}{3}\) and \(\frac{8}{3}\).
Answer: \(\frac{7}{3}\) and \(\frac{8}{3}\)

ii. \(\frac{1}{3}\) and \(\frac{2}{5}\)

Step 1: Find the L.C.M. of 3 and 5, which is 15. Convert both to like fractions.
\(\frac{1}{3} = \frac{1 \times 5}{15} = \frac{5}{15}\)
\(\frac{2}{5} = \frac{2 \times 3}{15} = \frac{6}{15}\)
Step 2: To find two numbers, multiply the numerator and denominator by 3.
\(\frac{5 \times 3}{15 \times 3} = \frac{15}{45}\) and \(\frac{6 \times 3}{15 \times 3} = \frac{18}{45}\).
Step 3: Numbers between \(\frac{15}{45}\) and \(\frac{18}{45}\) are \(\frac{16}{45}\) and \(\frac{17}{45}\).
Answer: \(\frac{16}{45}\) and \(\frac{17}{45}\)

iii. \(\frac{3}{4}\) and \(1\frac{1}{5}\)

Step 1: Convert mixed fraction to improper: \(1\frac{1}{5} = \frac{6}{5}\).
Step 2: L.C.M. of 4 and 5 is 20. Convert to like fractions.
\(\frac{3}{4} = \frac{15}{20}\) and \(\frac{6}{5} = \frac{24}{20}\).
Step 3: Any two numbers between \(\frac{15}{20}\) and \(\frac{24}{20}\) will work. Let’s take \(\frac{16}{20}\) and \(\frac{17}{20}\).
Step 4: Simplify \(\frac{16}{20} = \frac{4}{5}\).
Answer: \(\frac{4}{5}\) and \(\frac{17}{20}\)

iv. -2 and 1

Step 1: Since there are many integers between -2 and 1, we can simply pick any two rational numbers (integers are also rational).
Step 2: Numbers between -2 and 1 are -1 and 0.
Answer: -1 and 0

Q6: Find three rational numbers between:

i. 4 and 5

Step 1: To find 3 numbers, multiply by \(n+1 = 4\).
\(4 = \frac{4 \times 4}{4} = \frac{16}{4}\)
\(5 = \frac{5 \times 4}{4} = \frac{20}{4}\)
Step 2: Rational numbers between \(\frac{16}{4}\) and \(\frac{20}{4}\) are \(\frac{17}{4}, \frac{18}{4}, \frac{19}{4}\).
Answer: \(\frac{17}{4}, \frac{9}{2}, \frac{19}{4}\)

ii. \(\frac{1}{2}\) and \(\frac{3}{5}\)

Step 1: L.C.M. of 2 and 5 is 10. Convert to like fractions.
\(\frac{1}{2} = \frac{5}{10}\) and \(\frac{3}{5} = \frac{6}{10}\).
Step 2: Multiply both by 4: \(\frac{5 \times 4}{10 \times 4} = \frac{20}{40}\) and \(\frac{6 \times 4}{10 \times 4} = \frac{24}{40}\).
Step 3: Numbers are \(\frac{21}{40}, \frac{22}{40}, \frac{23}{40}\).
Answer: \(\frac{21}{40}, \frac{11}{20}, \frac{23}{40}\)

iii. -1 and 1

Step 1: We can pick simple fractions between -1 and 1.
Step 2: For example: \(-\frac{1}{2}, 0, \frac{1}{2}\).
Answer: \(-\frac{1}{2}, 0, \frac{1}{2}\)

iv. \(2\frac{1}{3}\) and \(3\frac{2}{3}\)

Step 1: Convert to improper fractions: \(\frac{7}{3}\) and \(\frac{11}{3}\).
Step 2: Numbers between \(\frac{7}{3}\) and \(\frac{11}{3}\) are \(\frac{8}{3}, \frac{9}{3}, \frac{10}{3}\).
Answer: \(\frac{8}{3}, 3, \frac{10}{3}\)

v. \(-\frac{1}{2}\) and \(\frac{1}{3}\)

Step 1: L.C.M. of 2 and 3 is 6. Convert to like fractions.
\(-\frac{1}{2} = -\frac{3}{6}\) and \(\frac{1}{3} = \frac{2}{6}\).
Step 2: Numbers between \(-\frac{3}{6}\) and \(\frac{2}{6}\) are \(-\frac{2}{6}, -\frac{1}{6}, 0\).
Answer: \(-\frac{1}{3}, -\frac{1}{6}, 0\)

vi. \(-\frac{1}{3}\) and \(\frac{1}{4}\)

Step 1: L.C.M. of 3 and 4 is 12. Convert to like fractions.
\(-\frac{1}{3} = -\frac{4}{12}\) and \(\frac{1}{4} = \frac{3}{12}\).
Step 2: Numbers between \(-\frac{4}{12}\) and \(\frac{3}{12}\) are \(-\frac{3}{12}, -\frac{2}{12}, -\frac{1}{12}\).
Answer: \(-\frac{1}{4}, -\frac{1}{6}, -\frac{1}{12}\)

Q7: Find four rational numbers between 4 and 4.5.

Step 1: Express 4 and 4.5 as fractions.
4 = \(\frac{4}{1}\) and 4.5 = \(\frac{45}{10} = \frac{9}{2}\).
Step 2: To find four numbers, we need a common denominator. Let’s use 10.
4 = \(\frac{4 \times 10}{10} = \frac{40}{10}\)
4.5 = \(\frac{45}{10}\)
Step 3: Since there are only 4 integers between 40 and 45 (41, 42, 43, 44), we can pick these for our numerators.
The numbers are \(\frac{41}{10}, \frac{42}{10}, \frac{43}{10}, \frac{44}{10}\).
Step 4: Simplify the fractions where possible:
\(\frac{41}{10} = 4.1\)
\(\frac{42}{10} = 4.2 = \frac{21}{5}\)
\(\frac{43}{10} = 4.3\)
\(\frac{44}{10} = 4.4 = \frac{22}{5}\)
Answer: 4.1, 4.2, 4.3, 4.4 (or \(\frac{41}{10}, \frac{21}{5}, \frac{43}{10}, \frac{22}{5}\))

Q8: Find six rational numbers between 3 and 4.

Step 1: To find 6 rational numbers, we multiply the numerator and denominator of both numbers by \(n+1 = 6+1 = 7\).
Step 2: Convert 3 into a fraction with denominator 7:
\(3 = \frac{3 \times 7}{7} = \frac{21}{7}\)
Step 3: Convert 4 into a fraction with denominator 7:
\(4 = \frac{4 \times 7}{7} = \frac{28}{7}\)
Step 4: Identify the six rational numbers between \(\frac{21}{7}\) and \(\frac{28}{7}\):
The numbers are \(\frac{22}{7}, \frac{23}{7}, \frac{24}{7}, \frac{25}{7}, \frac{26}{7}, \text{ and } \frac{27}{7}\).
Answer: \(\frac{22}{7}, \frac{23}{7}, \frac{24}{7}, \frac{25}{7}, \frac{26}{7}, \frac{27}{7}\)