Chapter 9 — Rational Numbers

What is a rational number?

A rational number is any number that can be written in the form p/q where p and q are integers and q ≠ 0. Here p is called the numerator and q the denominator. Examples: 3/4, −5/2, 7/1 (which is 7), and 0 (which can be written as 0/5).

Key points and properties

• Every integer is a rational number because an integer n = n/1.
• Rational numbers include positive fractions, negative fractions and zero.
• Decimal form: A rational number either has a terminating decimal expansion (e.g. 0.25 = 1/4) or a repeating (recurring) decimal (e.g. 0.333... = 1/3).
• Equivalent fractions: p/q = (p×k)/(q×k) for any nonzero integer k. We usually write a fraction in its simplest form where gcd(p,q)=1.
• Closure: Rational numbers are closed under addition, subtraction and multiplication. Division of two rationals is rational provided we do not divide by zero.
• Density: Between any two distinct rational numbers there exists another rational number (for example the average).

How to represent on the number line

To plot a fraction p/q on the number line: divide each unit segment into q equal parts and count p parts to the right if p is positive or to the left if p is negative. Example: to plot 2/3, divide 0–1 into 3 equal parts and go two parts to the right of 0.

Relation with irrational numbers

Numbers that cannot be written as a ratio of two integers (for example √2, π) are called irrational. Together, rational and irrational numbers make up the real numbers.

Rational numbers are numbers of the form a/b where a and b are integers and b ≠ 0. Two rational numbers a/b and c/d are equivalent if they represent the same number (same point on the number line). Algebraically, a/b and c/d are equivalent iff ad = bc.

You can generate equivalent rational numbers by multiplying (or dividing) both numerator and denominator by the same nonzero integer k: a/b = (a×k)/(b×k), k ≠ 0. Equivalent fractions look different but have the same value (for example 1/2 = 2/4 = 3/6).

Standard form (also called simplest form or lowest terms) of a rational number is the form in which the numerator and the denominator are coprime (their greatest common divisor is 1) and the denominator is positive. To convert a/b into standard form:

1. Find g = gcd(|a|, |b|).
2. Divide numerator and denominator by g: (a/g)/(b/g).
3. If the denominator is negative, multiply numerator and denominator by −1 so the denominator becomes positive.

Special cases: any integer n can be written as n/1 in standard form; 0 is represented as 0/1.

Example workflow: 42/56 → gcd(42,56)=14 → (42÷14)/(56÷14)=3/4, so 42/56 is equivalent to and has standard form 3/4. For signs: 6/−8 → multiply top and bottom by −1 to get −6/8 → gcd(6,8)=2 → −3/4.

Why this matters: working with equivalent forms and standard form makes it easy to compare, add or subtract rational numbers, and to recognize that different-looking fractions (or ratios) can represent the same real quantity in measurements, recipes, probabilities and map scales.

What is a number line? A number line is a straight horizontal line with a fixed point called the origin (labelled 0). Numbers to the right of 0 are positive and to the left are negative. Points equally spaced to the right/left mark the integers 1, 2, 3,... and -1, -2, -3,...

Rational numbers on the number line
Any rational number is of the form p/q (where p and q are integers and q ≠ 0). To represent p/q on the number line:

• Choose the origin 0 and mark the unit distance (point 1) to the right of 0.
• Divide the segment from 0 to 1 into q equal parts.
• If p/q is positive, count p parts to the right of 0; if p/q is negative, count p parts to the left of 0.

Improper fractions and mixed numbers
If p/q > 1 (an improper fraction), first mark the integers and then count the remaining fractional parts after the appropriate integer. For example 7/4 = 1 + 3/4, so locate 1 then move 3 parts of the quarter-unit further.

Equivalent rational numbers
Fractions that are equal (for example 1/2 and 2/4) fall on the same point. You can create equivalent fractions by multiplying numerator and denominator by the same nonzero integer.

Ordering and comparison
The number line gives a visual method to compare rational numbers: the number located to the right is larger. For negative numbers, a point closer to 0 (less to the left) is larger.

Density of rationals
Between any two distinct rational numbers there are infinitely many rational numbers. A practical way to find one is to take the midpoint: for a/b and c/d, midpoint = (a/b + c/d)/2 (which is also rational).

Distance between two rational numbers
The distance (absolute difference) between a/b and c/d on the number line is |a/b - c/d|; this gives how far apart their points are.

What is a rational number? A rational number is any number that can be expressed as p/q where p and q are integers and q ≠ 0. Examples: 2/3, -5/4, 7, 0 (7 = 7/1, 0 = 0/1).

Basic idea of operations: We perform addition, subtraction, multiplication and division on rational numbers using rules similar to whole-number arithmetic but taking care of denominators and signs. Fractions should be simplified (reduced) to lowest terms when possible.

Addition and subtraction

• If denominators are equal: add/subtract numerators and keep the denominator. Example: 3/8 + 2/8 = (3+2)/8 = 5/8.
• If denominators are different: convert to equivalent fractions with a common denominator (usually LCM), then add/subtract numerators. Example: 2/3 + 3/4 = LCM(3,4)=12 → 8/12 + 9/12 = 17/12 = 1 5/12.
• Subtraction can be treated as addition of the additive inverse: a/b − c/d = a/b + (−c/d).

Multiplication

• Multiply numerators to get the new numerator and denominators to get the new denominator: (a/b)×(c/d) = (ac)/(bd). Simplify afterwards. Example: (−2/5)×(15/4) = (−30)/(20) = −3/2 after dividing numerator and denominator by 10.
• It is often helpful to cancel common factors between a numerator and the other fraction's denominator before multiplying.

Division

• To divide by a nonzero rational number, multiply by its reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c), where c ≠ 0.
• Example: (−3/7) ÷ (6/5) = (−3/7)×(5/6) = (−15)/(42) = (−5)/14 after simplification.

Rules for signs

• Positive × positive = positive
• Negative × negative = positive
• Positive × negative = negative (same for division)
• Adding numbers with different signs: subtract smaller absolute value from larger absolute value and keep sign of the larger absolute value.

Properties

• Commutative: a/b + c/d = c/d + a/b and (a/b)×(c/d) = (c/d)×(a/b)
• Associative: (a/b + c/d) + e/f = a/b + (c/d + e/f) and similarly for multiplication
• Distributive: (a/b)×(c/d + e/f) = (a/b)×(c/d) + (a/b)×(e/f)
• Additive identity: a/b + 0 = a/b. Multiplicative identity: a/b × 1 = a/b.

Worked examples

1. Addition: 2/3 + 3/4 = LCM(3,4)=12 → 8/12 + 9/12 = 17/12 = 1 5/12.
2. Subtraction: 5/6 − 3/4 = LCM(6,4)=12 → 10/12 − 9/12 = 1/12.
3. Multiplication: (−2/5) × (15/4) = cancel 5 with 15 → (−2/1)×(3/4) = (−6)/4 = (−3)/2.
4. Division: (−3/7) ÷ (6/5) = (−3/7) × (5/6) = (−15)/42 = (−5)/14.
5. Distributive check: (1/2)×(2/3 + 3/4) = (1/2)×(17/12) = 17/24. Using distribution: (1/2×2/3)+(1/2×3/4) = 1/3 + 3/8 = 8/24 + 9/24 = 17/24.

Tips: Always simplify fractions by dividing numerator and denominator by their GCD. When adding/subtracting, find the LCM of denominators. For multiplication/division, cancel common factors early to keep numbers small.

What is a rational number? A rational number is any number that can be expressed as p/q where p and q are integers and q ≠ 0. Examples: 3/4, -5, 0, 0.75 (which is 3/4), 2 (which is 2/1).

Main properties

• Closure: Rational numbers are closed under addition, subtraction and multiplication. That means if a and b are rational, then a+b, a−b and a·b are rational. (Not closed under division by zero.)
• Commutative property: For any rationals a and b,
  a + b = b + a, and a · b = b · a.
• Associative property: For any rationals a, b, c,
  (a + b) + c = a + (b + c), and (a · b) · c = a · (b · c).
• Distributive property: Multiplication distributes over addition:
  a · (b + c) = a · b + a · c.
• Identity elements: 0 is the additive identity because a + 0 = a. 1 is the multiplicative identity because a · 1 = a.
• Inverse elements:
  • Additive inverse: For every rational a, there exists −a such that a + (−a) = 0.
  • Multiplicative inverse: For every nonzero rational a, there exists 1/a such that a · (1/a) = 1.
• Density: Between any two distinct rational numbers there is another rational number. For example, the average (a + b)/2 is rational and lies between a and b.
• Decimal representation: Every rational number expressed in decimal form either terminates (e.g., 1/4 = 0.25) or repeats periodically (e.g., 1/3 = 0.333...).
• Representation on number line: Rational numbers can be located exactly on the number line by marking fractions or equivalent fractions (e.g., 1/2 = 2/4).

Remarks: Division of a rational number by 0 is undefined, so the set of rational numbers is not closed under division by zero. All integers are rational (n = n/1), but not all real numbers are rational (e.g., √2 is irrational).

What it means: Rational numbers are numbers that can be written as p/q where p and q are integers and q ≠ 0. The density property of rational numbers says: between any two distinct rational numbers there is always another rational number — in fact, there are infinitely many rational numbers between them.

Simple proof (one number between any two): Let a and b be rational with a < b. The midpoint (a + b)/2 is rational and satisfies a < (a + b)/2 < b. Hence at least one rational number lies between a and b.

More: infinitely many between any two: To get many rationals between a and b, choose any positive integer n. Then the n numbers
(a + (b-a)/(n+1)), (a + 2(b-a)/(n+1)), ..., (a + n(b-a)/(n+1))
are all rational and lie strictly between a and b. Since n can be any positive integer, there are infinitely many rationals between a and b.

Alternate constructive idea: If a = p/q and b = r/s, you can put them over a common denominator q·s: a = (p·s)/(q·s), b = (r·q)/(q·s). If the numerators differ by more than 1, you can choose an integer between them to create a new rational. If not, multiply numerator and denominator of both fractions by a suitable integer (e.g. 10, 100, ...) until there is room for integers between the new numerators. This shows you can explicitly construct many rationals between any two given ones.

Intuition: Rational numbers are dense on the number line — no matter how close two rational numbers are, you can always find another rational between them. Visually this is like being able to keep zooming in between two points and always finding more points with rational coordinates.

What it means: A decimal representation is another way to write a rational number (a number of the form p/q where p and q are integers and q ≠ 0). We get the decimal form by dividing the numerator by the denominator (long division).

Two kinds of decimal forms:

• Terminating decimal: the division ends after a finite number of digits (example: 7/8 = 0.875).
• Recurring (repeating) decimal: after some point digits repeat forever (example: 1/3 = 0.3, 22/7 = 3.142857).

Key test for termination: Reduce p/q to simplest form. If the denominator (after simplification) has only 2 and/or 5 as prime factors (i.e. q = 2^a 5^b), the decimal will terminate. Otherwise it will be recurring.

How to convert a fraction to a decimal:

1. Perform long division p ÷ q. If division ends, decimal is terminating.
2. If division begins repeating a remainder, the digits from the first repeated remainder repeat forever → recurring decimal.
3. For terminating decimals you may also multiply numerator and denominator by a number to make denominator 10^k, then read off the decimal.

How to convert a decimal to a fraction:

• Terminating decimal: Put the digits (without the point) over 10^k where k is number of digits after decimal, then simplify. Example: 0.375 = 375/1000 = 3/8.
• Pure repeating decimal (all digits after decimal repeat): If x = 0.d and the repeating block has r digits, then 10^r x - x = integer -> x = repeating_digits / (10^r - 1). Example: 0.36 = 36/99 = 4/11.
• Mixed (non‑repeating prefix + repeating block): If n digits do not repeat and r digits repeat, set up 10^{n+r}x and 10^n x and subtract: (10^{n+r}x - 10^n x) = integer formed by all digits up to the end of first repeat minus digits of non‑repeating part. Denominator = 10^{n+r}-10^n. Simplify. Example method given below.

Notation: We mark repeating digits with a bar, e.g., 0.27 = 0.272727... .

Practical tips:

• Always simplify the fraction before testing denominator factors.
• When using long division, keep track of remainders: repeating remainder => repeating decimal, first occurrence of remainder 0 => terminating decimal.
• For converting mixed repeating decimals, choose powers of 10 so that repeated portion lines up when subtracting.

What the topic covers
Applications and Word Problems involve translating real-life situations into expressions and equations with rational numbers (integers, fractions and terminating/repeating decimals) and then solving them using the rules of operations on rational numbers.

How to approach word problems

• Read carefully: Identify quantities, units and what is asked.
• Assign signs: Use positive for gains/above/forward/right, negative for losses/below/backward/left.
• Convert: Express all quantities as rational numbers (convert mixed numbers to improper fractions if needed).
• Choose operations: Decide whether to add, subtract, multiply or divide (or combine operations).
• Solve stepwise: Use rules for fractions/decimals and simplify the result.
• Interpret: Give the answer with correct units and sign, and check if it makes sense.

Common techniques used

• Use number-line models for additions and subtractions (helps to visualise sign changes).
• Use bar models or fractional-parts diagrams for sharing, mixture and part–whole problems.
• Convert to common denominators (LCM) for addition/subtraction of fractions.
• Convert mixed numbers to improper fractions for multiplication/division.
• Use reciprocal to divide by a fraction.

Practice and Exercises for Rational Numbers help students master identification, representation and arithmetic operations with rational numbers (fractions and integers expressed as a/b where a and b are integers and b ≠ 0). Exercises focus on: converting between mixed and improper fractions, ordering and comparing rational numbers, plotting on the number line, simplifying, and performing addition, subtraction, multiplication and division using rules and properties.

Key strategies used in exercises:

• Convert mixed numbers to improper fractions before operations and back to mixed form if needed.
• For addition and subtraction, always find the least common multiple (LCM) of denominators to get a common denominator; then add/subtract numerators and simplify the result.
• For multiplication, multiply numerators and denominators; simplify by cancelling common factors before multiplying to keep numbers small.
• For division, multiply by the reciprocal of the divisor: a/b ÷ c/d = a/b × d/c (provided c ≠ 0).
• Use sign rules: product or quotient of two numbers with the same sign is positive; with opposite signs is negative. Zero divided by any nonzero number is 0; division by zero is undefined.

Typical exercise types include: straightforward computations, simplification, conversion between mixed and improper forms, ordering a list of rational numbers, plotting and interpreting points on a number line, and word problems (money, temperature, distances, fractions of quantities). Regular practice improves fluency in finding common denominators, cancelling factors and applying sign rules correctly.

Errors to watch for: forgetting to simplify, neglecting to convert mixed numbers, not cancelling common factors before multiplying, and sign mistakes in subtraction/division. Use the number line to check results visually: adding a positive moves right, adding a negative moves left.