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A Story of Numbers - Reema’s Curiosity

Grade 8CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Introduction to the number system: Natural numbers ({1,2,3...}\{1, 2, 3...\}), Whole numbers ({0,1,2...}\{0, 1, 2...\}), and Integers ({...,−2,−1,0,1,2...}\{..., -2, -1, 0, 1, 2...\}).

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Rational Numbers: Any number that can be expressed in the form pq\frac{p}{q}, where pp and qq are integers and q≠0q \neq 0.

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Closure Property: For any two rational numbers aa and bb, a+ba+b, a−ba-b, and a×ba \times b are also rational numbers. Division is closed only if we exclude zero.

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Commutativity: For rational numbers, addition and multiplication are commutative (a+b=b+aa + b = b + a and a×b=b×aa \times b = b \times a), but subtraction and division are not.

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Associativity: Addition and multiplication are associative for rational numbers, such that a+(b+c)=(a+b)+ca + (b + c) = (a + b) + c.

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Role of Zero and One: 00 is the additive identity (a+0=aa + 0 = a) and 11 is the multiplicative identity (a×1=aa \times 1 = a).

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Negative of a number: For a rational number ab\frac{a}{b}, its additive inverse is −ab-\frac{a}{b} because ab+(−ab)=0\frac{a}{b} + (-\frac{a}{b}) = 0.

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Reciprocal: For a non-zero rational number ab\frac{a}{b}, its multiplicative inverse is ba\frac{b}{a} because ab×ba=1\frac{a}{b} \times \frac{b}{a} = 1.

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Distributivity: Multiplication is distributive over addition and subtraction: a(b+c)=ab+aca(b + c) = ab + ac and a(b−c)=ab−aca(b - c) = ab - ac.

📐Formulae

Rational Number=pq,q≠0\text{Rational Number} = \frac{p}{q}, q \neq 0

a+b=b+a (Commutative Property of Addition)a + b = b + a \text{ (Commutative Property of Addition)}

a×b=b×a (Commutative Property of Multiplication)a \times b = b \times a \text{ (Commutative Property of Multiplication)}

a+(b+c)=(a+b)+c (Associative Property of Addition)a + (b + c) = (a + b) + c \text{ (Associative Property of Addition)}

a×(b×c)=(a×b)×c (Associative Property of Multiplication)a \times (b \times c) = (a \times b) \times c \text{ (Associative Property of Multiplication)}

a(b+c)=ab+ac (Distributive Property)a(b + c) = ab + ac \text{ (Distributive Property)}

Additive Inverse: a+(−a)=0\text{Additive Inverse: } a + (-a) = 0

Multiplicative Inverse: a×1a=1\text{Multiplicative Inverse: } a \times \frac{1}{a} = 1

💡Examples

Problem 1:

Find the additive inverse of −719\frac{-7}{19} and the multiplicative inverse of 58\frac{5}{8}.

Solution:

Additive inverse is 719\frac{7}{19}. Multiplicative inverse is 85\frac{8}{5}.

Explanation:

The additive inverse of a number xx is −x-x such that x+(−x)=0x + (-x) = 0. So, −(−719)=719-\left(\frac{-7}{19}\right) = \frac{7}{19}. The multiplicative inverse is the reciprocal, such that 58×85=1\frac{5}{8} \times \frac{8}{5} = 1.

Problem 2:

Simplify using distributive property: 25×[−37]−114−37×35\frac{2}{5} \times \left[ \frac{-3}{7} \right] - \frac{1}{14} - \frac{3}{7} \times \frac{3}{5}

Solution:

−12\frac{-1}{2}

Explanation:

Rearrange the terms: 25×(−37)−37×35−114\frac{2}{5} \times \left( \frac{-3}{7} \right) - \frac{3}{7} \times \frac{3}{5} - \frac{1}{14}. Take −37\frac{-3}{7} common using distributivity: −37(25+35)−114=−37(1)−114=−6−114=−714=−12\frac{-3}{7} \left( \frac{2}{5} + \frac{3}{5} \right) - \frac{1}{14} = \frac{-3}{7}(1) - \frac{1}{14} = \frac{-6-1}{14} = \frac{-7}{14} = \frac{-1}{2}.

Problem 3:

Calculate the difference between 80,000,00080,000,000 and 34,567,89234,567,892 to show the property of subtraction in large integers.

Solution:

80000000−3456789245432108\begin{array}{r} 80000000 \\ -34567892 \\ \hline 45432108 \end{array}

Explanation:

Performing vertical subtraction by borrowing across zeros, we find the result is 45,432,10845,432,108, which is also an integer, demonstrating the closure property of subtraction for integers.