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Rational Numbers - Properties of Rational Numbers

Grade 8ICSE

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

🔑Concepts

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Definition of Rational Numbers: A rational number is any number that can be expressed in the form pq\frac{p}{q}, where pp and qq are integers and q≠0q \neq 0. Visually, these numbers occupy specific points on a number line between integers, representing parts of a whole.

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Closure Property: Rational numbers are closed under addition, subtraction, and multiplication. This means the result of these operations on any two rational numbers is always another rational number. However, they are not closed under division if the divisor is zero.

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Commutative Property: For any two rational numbers aa and bb, addition and multiplication are commutative (a+b=b+aa + b = b + a and a×b=b×aa \times b = b \times a). Subtraction and division do not follow this property. Imagine a balance scale; swapping the order of weights in addition keeps the scale balanced.

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Associative Property: For any three rational numbers aa, bb, and cc, the grouping of numbers does not change the result for addition and multiplication: (a+b)+c=a+(b+c)(a + b) + c = a + (b + c) and (a×b)×c=a×(b×c)(a \times b) \times c = a \times (b \times c).

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Distributive Property: Multiplication of rational numbers is distributive over addition and subtraction: a×(b+c)=(a×b)+(a×c)a \times (b + c) = (a \times b) + (a \times c). Geometrically, this is like finding the area of a large rectangle by splitting it into two smaller rectangles and adding their areas.

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Identity Elements: Zero (00) is the additive identity because adding it to any rational number leaves the number unchanged (a+0=aa + 0 = a). One (11) is the multiplicative identity because multiplying any rational number by it leaves the number unchanged (a×1=aa \times 1 = a).

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Inverse Elements: Every rational number ab\frac{a}{b} has an additive inverse −ab-\frac{a}{b} such that their sum is 00. Every non-zero rational number ab\frac{a}{b} has a multiplicative inverse (reciprocal) ba\frac{b}{a} such that their product is 11.

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Density Property: Between any two rational numbers, there are infinitely many rational numbers. This can be visualized by repeatedly magnifying the space between two points on a number line, revealing more and more fractional points.

📐Formulae

General Form: Q={pq:p,q∈Z,q≠0}Q = \{\frac{p}{q} : p, q \in Z, q \neq 0\}

Additive Identity: a+0=0+a=aa + 0 = 0 + a = a

Multiplicative Identity: a×1=1×a=aa \times 1 = 1 \times a = a

Additive Inverse: a+(−a)=0a + (-a) = 0

Multiplicative Inverse: a×1a=1a \times \frac{1}{a} = 1 (where a≠0a \neq 0)

Distributive Law: a(b±c)=ab±aca(b \pm c) = ab \pm ac

Mean Method (to find a number between aa and bb): a+b2\frac{a + b}{2}

💡Examples

Problem 1:

Simplify using suitable properties: 35×(−27)−110−35×57\frac{3}{5} \times \left(-\frac{2}{7}\right) - \frac{1}{10} - \frac{3}{5} \times \frac{5}{7}

Solution:

Step 1: Group the terms with the common factor 35\frac{3}{5} using the Commutative property: 35×(−27)−35×57−110\frac{3}{5} \times \left(-\frac{2}{7}\right) - \frac{3}{5} \times \frac{5}{7} - \frac{1}{10} Step 2: Apply the Distributive property a×b+a×c=a(b+c)a \times b + a \times c = a(b + c): 35×[−27−57]−110\frac{3}{5} \times \left[-\frac{2}{7} - \frac{5}{7}\right] - \frac{1}{10} Step 3: Solve inside the brackets: 35×[−77]−110=35×(−1)−110\frac{3}{5} \times \left[-\frac{7}{7}\right] - \frac{1}{10} = \frac{3}{5} \times (-1) - \frac{1}{10} Step 4: Perform multiplication: −35−110-\frac{3}{5} - \frac{1}{10} Step 5: Find a common denominator (1010): −610−110=−710-\frac{6}{10} - \frac{1}{10} = -\frac{7}{10}

Explanation:

This problem uses the Commutative property to rearrange terms and the Distributive property to simplify the calculation by factoring out the common rational number.

Problem 2:

Find two rational numbers between 13\frac{1}{3} and 12\frac{1}{2}.

Solution:

Step 1: Make the denominators the same. The LCM of 33 and 22 is 66. 13=1×23×2=26\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6} 12=1×32×3=36\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6} Step 2: Since there are no integers between the numerators 22 and 33, multiply both by a larger number like 33: 2×36×3=618\frac{2 \times 3}{6 \times 3} = \frac{6}{18} 3×36×3=918\frac{3 \times 3}{6 \times 3} = \frac{9}{18} Step 3: Identify numbers between 618\frac{6}{18} and 918\frac{9}{18}: The numbers are 718\frac{7}{18} and 818\frac{8}{18} (which simplifies to 49\frac{4}{9}).

Explanation:

This approach uses the property of equivalent fractions to expand the gap between two rational numbers, making it easy to identify intermediate values.