Real and Complex Numbers - Exercise 1.2 - Mathematics 9th

Mathematics 9th - Exercise 1.2

REAL AND COMPLEX NUMBERS

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Question 1:

Recognize the properties of real numbers used in the following:

(i)

$$\frac{1}{2} + \frac{2}{3} = \frac{2}{3} + \frac{1}{2}$$

Solution:

$$a + b = b + a$$

Commutative property of addition

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(ii)

$$\frac{4}{3} + \left( \frac{1}{3} + \frac{2}{3} \right) = \left( \frac{4}{3} + \frac{1}{3} \right) + \frac{2}{3}$$

Solution:

$$a + (b + c) = (a + b) + c$$

Associative property of addition

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(iii)

$$9 \times \left( \frac{10}{9} + \frac{20}{9} \right) = \left( 9 \times \frac{10}{9} \right) + \left( 9 \times \frac{20}{9} \right)$$

Solution:
Left distributive property of multiplication over addition

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(iv)

$$\left( \frac{4}{5} + \frac{7}{8} \right) \times \frac{5}{7} = \left( \frac{4}{5} \times \frac{5}{7} \right) + \left( \frac{7}{8} \times \frac{5}{7} \right)$$

Solution:
Right distributive property of multiplication over addition

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(v)

$$\left( \frac{7}{5} - \frac{3}{5} \right) \times \frac{10}{15} = \left( \frac{7}{5} \times \frac{10}{15} \right) - \left( \frac{3}{5} \times \frac{10}{15} \right)$$

Solution:
Right distributive property of multiplication over subtraction

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(vi)

$$\frac{d}{c} \times \frac{e}{f} = \frac{e}{f} \times \frac{d}{c}$$

Solution:
Commutative property of multiplication

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(vii)

$$11 \times (15 \times 21) = (11 \times 15) \times 21$$

Solution:
Associative property of multiplication

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(viii)

$$2 \div 11 \times 11 \div 2 = 1$$

Solution:
Multiplicative inverse

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(ix)

$$\left( \frac{3}{5} \right) + \left( -\frac{3}{5} \right) = 0$$

Solution:
Additive inverse

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(x)

$$\left( \frac{a}{b} \times \frac{b}{a} \right) = 1$$

Solution:
Multiplicative inverse

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(xi)

$$\left( \frac{15}{10} \times \frac{8}{5} - \frac{4}{10} \right) = \left( \frac{15}{10} \times \frac{8}{5} \right) - \left( \frac{15}{10} \times \frac{4}{10} \right)$$

Solution:
Left distributive property of multiplication over subtraction

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(xii)

$$\frac{\sqrt{2}}{3} \times \frac{3}{\sqrt{2}} = 1$$

Solution:
Multiplicative inverse

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Question 2:

Fill the correct real number in the following to make the real numbers property correct.

(i)

$$\frac{\sqrt{2}}{5} + \frac{3}{\sqrt{6}} + \square = \frac{3}{\sqrt{6}} + \square + \frac{\sqrt{2}}{5}$$

Solution:

$$a + b = b + a$$

Answer: $\boxed{\frac{\sqrt{2}}{5}}$

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(ii)

$$7 \div 10 + \left( 70 \div 16 \div 33 \right) = \square + \left( 7 \div 10 \right)$$

Solution:

$$a + (b + c) = c + (a + b)$$

Answer: $\boxed{70 \div 16 \div 33}$

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(iii)

$$\frac{99}{50} \times \frac{50}{99} = \square$$

Solution:
Answer: $\boxed{1}$

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(iv)

$$\frac{59}{95} \times \frac{95}{59} = \square$$

Solution:
Answer: $\boxed{1}$

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(v)

$$(-21) + \square = 0$$

Solution:
Answer: $\boxed{21}$

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(vi)

$$\left( \frac{5}{8} \times \frac{2}{3} \right) - \square = \left( \frac{5}{8} \times \frac{2}{3} \right) + \left( \frac{5}{8} \times \frac{5}{7} \right)$$

Solution:
Answer: $\boxed{5/7}$

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Question 3:

Fill the following blanks to make the property correct/true:

1. $5 < 8$ and $8 < 10 \Rightarrow 5 < \boxed{10}$
2. $10 > 8$ and $8 > 5 \Rightarrow 10 > \boxed{5}$
3. $3 < 6 \Rightarrow 3 + 9 < \boxed{6 + 9}$
4. $4 < 6 \Rightarrow 4 + 8 < \boxed{6 + 8}$
5. $8 > 6 \Rightarrow 6 + 8 < \boxed{6 + 6}$

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Question 4:

Fill the following blanks to make the property correct/true:

1. $5 < 7 \Rightarrow 5 \times 12 < \boxed{7 \times 12}$
2. $7 > 5 \Rightarrow 7 \times 12 > \boxed{5 \times 12}$
3. $6 > 4 \Rightarrow 6 \times (-7) < \boxed{4 \times (-7)}$
4. $2 < 8 \Rightarrow 2 \times (-4) > \boxed{8 \times (-4)}$
5. $8 > 6 \Rightarrow 6 + 8 < \boxed{6 + 6}$

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Question 5:

Find the additive and multiplicative inverse of the following real numbers:

(i) 3

• Additive inverse: $3 + (-3) = 0$
• Multiplicative inverse: $3 \times \frac{1}{3} = 1$

(ii) -7

• Additive inverse: $-7 + 7 = 0$
• Multiplicative inverse: $-7 \times \frac{1}{-7} = 1$

(iii) 0.3

• Additive inverse: $0.3 + (-0.3) = 0$
• Multiplicative inverse: $0.3 \times \frac{1}{0.3} = 1$

(iv)

$$-\frac{\sqrt{5}}{5}$$

• Additive inverse: $-\frac{\sqrt{5}}{5} + \frac{\sqrt{5}}{5} = 0$
• Multiplicative inverse:

$$-\frac{\sqrt{5}}{5} \times \left(-\frac{5}{\sqrt{5}}\right) = 1$$

(v)

$$\frac{9}{\sqrt{\mathrm{12}}}$$

• Additive inverse:

$$\frac{9}{\sqrt{12}}+(-\frac{9}{\sqrt{12}})=0$$

• Multiplicative inverse:

$$\frac{9}{\sqrt{12}}\times \frac{\sqrt{12}}{9}=1$$

(vi) 0

• Additive inverse: $0 + 0 = 0$