Introduction
Numbers have a rich story — from natural numbers to integers, fractions, decimals, and rational numbers. This chapter explores the vast world of rational numbers, their properties, and how they fit into the bigger picture of the number line.
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Key Concepts
Rational numbers: Numbers that can be written in the form p/q, where p and q are integers and q is not equal to 0. Examples: 3/4, -5/7, 0, -2, 1/2.
- Properties of Rational Numbers:
- Closure: Rational numbers are closed under addition, subtraction, multiplication. Division is closed except by 0.
- Commutativity: Addition and multiplication of rational numbers are commutative: a + b = b + a; a x b = b x a.
- Associativity: (a + b) + c = a + (b + c) and (a x b) x c = a x (b x c).
- Distributivity: a x (b + c) = a x b + a x c.
- Additive identity: 0. Multiplicative identity: 1.
- Additive inverse of p/q is -p/q. Multiplicative inverse (reciprocal) of p/q (not zero) is q/p.
Representation on the number line: Every rational number has a unique position on the number line. Positive rationals are to the right of 0; negative rationals are to the left.
Finding rational numbers between two rationals: Between any two rational numbers, there are infinitely many rational numbers. Method: use the mean — (a + b)/2 gives a rational number between a and b.
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Worked Examples
Verify commutativity of addition for 2/3 and 1/5.
2/3 + 1/5 = 10/15 + 3/15 = 13/15.
1/5 + 2/3 = 3/15 + 10/15 = 13/15. Both are equal, confirming commutativity.
Find the additive inverse of -4/9.
The additive inverse is 4/9, since (-4/9) + (4/9) = 0.
Find the multiplicative inverse of -7/3.
The multiplicative inverse (reciprocal) is -3/7, since (-7/3) x (-3/7) = 1.
Using distributivity, simplify: 3/4 x (8/9 + (-4/3)).
= (3/4 x 8/9) + (3/4 x (-4/3))
= 24/36 + (-12/12)
= 2/3 + (-1)
= 2/3 - 3/3 = -1/3.
Find two rational numbers between 1/4 and 1/2.
Mean of 1/4 and 1/2 = (1/4 + 1/2)/2 = (3/4)/2 = 3/8.
Mean of 3/8 and 1/2 = (3/8 + 4/8)/2 = (7/8)/2 = 7/16.
Two rational numbers: 3/8 and 7/16.
Represent -3/5 on a number line.
Divide the segment from -1 to 0 into 5 equal parts. -3/5 is 3 parts to the left of 0.
Show that 0 is neither positive nor negative rational number.
0 = 0/1. It has no sign, does not lie on either side of the number line; it is the origin.
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Common mistakes
- Confusing -p/q with p/(-q) — both are equal and represent the same negative rational.
- Thinking the reciprocal of 0 exists — it does not, since division by zero is undefined.
- Forgetting to find a common denominator before adding or subtracting rational numbers.
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Summary
Rational numbers extend integers by including all fractions. They obey closure, commutativity, associativity, and distributivity. Every rational number can be placed on the number line, and between any two rationals, infinitely many others exist. These properties make rational numbers a complete and powerful number system for arithmetic.