Chapter 1 — Rational Numbers

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

Key ideas

• Positive, negative and zero: If p and q have the same sign the rational is positive; if they have opposite signs it is negative; 0 is rational because 0 = 0/q.
• Equivalent rational numbers: p/q = (kp)/(kq) for any nonzero integer k. For example, 1/2 = 2/4 = 3/6.
• Proper and improper: Proper fraction: |p| < |q| (e.g., 3/4). Improper fraction: |p| ≥ |q| (e.g., 7/4). Improper can be written as a mixed number.
• Simplest form: A fraction is in simplest form when p and q have no common factor other than 1.

Representation on number line

To represent p/q: divide one unit interval into q equal parts and move p parts to the right (if p positive) or to the left (if p negative). Equivalent fractions occupy the same point on the number line.

Decimal representation

• A rational number has a decimal expansion that either terminates (e.g., 3/4 = 0.75) or repeats periodically (e.g., 1/3 = 0.333...).
• To convert a terminating decimal to a fraction: multiply to remove the decimal point and simplify (e.g., 0.125 = 125/1000 = 1/8).
• To convert a repeating decimal: use algebraic method (e.g., x = 0.777... ⇒ 10x - x = 7 ⇒ x = 7/9).

Important properties

• Closure: Sum and product of two rationals is rational.
• Commutative: a + b = b + a, a × b = b × a.
• Associative: (a + b) + c = a + (b + c), (a × b) × c = a × (b × c).
• Distributive: a(b + c) = ab + ac.
• Identities: Additive identity 0 (a + 0 = a); multiplicative identity 1 (a × 1 = a).
• Inverses: Additive inverse of a is -a (a + (-a) = 0); multiplicative inverse of nonzero a is 1/a (a × 1/a = 1).

Density property

Between any two distinct rational numbers there exists another rational number (for example, the midpoint (a + b)/2). This means rationals are dense on the number line.

Why this matters (real life)

Rational numbers model any quantity that can be expressed as a ratio of whole items: portions of food, money divided among people, measuring lengths, probabilities, concentrations, speed as ratio of distance/time, and many day-to-day fractions.

What are Rational Numbers? A rational number is any number that can be expressed as p/q where p and q are integers and q ≠ 0.

Standard Form: A rational number is said to be in its standard (or simplest) form when the numerator and denominator have no common factor other than 1 (i.e. they are coprime) and the denominator is positive. In other words, p/q is standard if gcd(p, q) = 1 and q > 0.

Why require denominator positive? To have a unique standard representation. A negative rational number should be written with the negative sign on the numerator or in front of the fraction: -a/b (with b > 0).

Equivalent Forms: Two fractions are equivalent if they represent the same rational number. If a/b is a rational number and k is any nonzero integer, then (ka)/(kb) represents the same rational number as a/b. Conversely, if a and b have a common factor d, dividing both by d gives an equivalent, simpler form.

How to get standard form (step-by-step):

• Find gcd(|p|, |q|).
• Divide numerator and denominator by the gcd.
• If denominator is negative, multiply numerator and denominator by -1 so denominator becomes positive.

Signs and placement: -a/b = a/-b = -(a/b). Standard form uses -a/b with b > 0.

Improper fraction and mixed number: An improper fraction (numerator ≥ denominator) can be written as a mixed number; conversion formulas are given below.

Decimals: After reducing to standard form, if the denominator (in prime factors) contains only 2s and/or 5s, the decimal expansion terminates; otherwise it is a repeating decimal. This helps identify equivalent decimal forms.

Summary: Equivalent forms are useful for comparing, adding, or scaling fractions; the standard form gives a unique simplified representation.

What is a number line? A number line is a straight horizontal line with a fixed origin labeled 0, equal-sized ticks to the right for positive numbers and to the left for negative numbers. It is used to represent rational numbers (fractions and integers) as points on the line.

Rules to represent a rational number a/b (b > 0):

1. Sign: If a > 0, move to the right of 0; if a < 0, move to the left; if a = 0, the point is at the origin.
2. Unit division: Consider the unit segment from 0 to 1 (or 0 to −1 for negative). Divide that unit into b equal parts.
3. Counting parts: From 0, count |a| of those equal parts toward the right (for positive) or toward the left (for negative). The point reached is a/b.
4. Improper fractions: Convert an improper fraction a/b into a mixed number q + r/b (where q is integer part and 0 ≤ r < b). First go to q (an integer tick) then count r parts into the next unit.

Important observations: Equivalent fractions (for example 1/2, 2/4, 3/6) represent the same point because they divide the unit into proportional parts. Rational numbers are dense: between any two distinct rational numbers there exists another rational number (so there are infinitely many rational points between any two points).

Use in comparisons: On the number line a point to the right is greater. To compare a/b and c/d (with bd > 0) you can cross-multiply: a/b < c/d iff ad < bc. The number line gives a visual check: the smaller (left) point is the smaller number.

What is decimal representation? A decimal representation is a way of writing rational numbers (fractions) using the decimal (base–10) place-value system. Every rational number can be expressed either as a terminating (finite) decimal or as a repeating (recurring) decimal.

Two types

• Terminating decimals: The decimal expansion ends after a finite number of digits (for example, 0.25, 0.375).
• Repeating (recurring) decimals: After some point a block of digits repeats forever (for example, 0.333..., 0.083333..., 0.142857142857...).

How to get a decimal from a fraction

1. Use long division: divide the numerator by the denominator. If remainder becomes zero, the decimal terminates; if a remainder repeats, the digits between repeats form the repeating block.
2. Termination test: A fraction in lowest terms a/b produces a terminating decimal iff the denominator b has no prime factors other than 2 and 5. In other words, b = 2^m * 5^n for some nonnegative integers m,n.

How to convert a terminating decimal to a fraction

If x has k decimal places, x = N / 10^k (then simplify the fraction). Example: 0.375 = 375/1000 = 3/8 after simplification.

How to convert a repeating decimal to a fraction (algebraic method)

For a purely repeating decimal with r repeating digits, e.g. x = 0.̅{d1d2...dr}, multiply by 10^r and subtract:

10^r x - x = integer formed by repeating block, so x = (repeating block)/ (10^r - 1).

For a decimal with s non-repeating digits followed by r repeating digits (example 0.a1a2...as b1b2...br repeating), multiply by 10^{s+r} and 10^s and subtract:

x = (integer formed by non-repeat+repeat - integer formed by non-repeat) / (10^{s+r} - 10^s).

Important consequences

• Every rational number has a decimal expansion that either terminates or eventually repeats.
• Decimals like 0.999... equal 1 (because they are repeating decimals corresponding to a rational number).

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

Representation: Rational numbers can be proper fractions, improper fractions, mixed numbers, or integers. Always simplify fractions to their lowest terms by dividing numerator and denominator by their HCF.

1. Addition and subtraction

• To add or subtract fractions, first write them with a common denominator (preferably the LCM of denominators).
• Formula (for unlike denominators): a/b + c/d = (ad + bc) / bd. Better: rewrite using LCM: a/b + c/d = (a*(LCM/b) + c*(LCM/d)) / LCM.
• For subtraction: a/b − c/d = (ad − bc) / bd (or use LCM method). If signs differ, convert to addition of the negative and compare magnitudes to decide sign.
• Example method: to compute 2/3 + (−5/6), LCM(3,6)=6 so 2/3 = 4/6; 4/6 + (−5/6) = −1/6.

2. Multiplication

• Multiply numerators and denominators directly: (a/b) × (c/d) = (a×c) / (b×d).
• Always cancel common factors between any numerator and any denominator before multiplying to simplify calculation.
• Sign rule: if factors have same sign result is positive; if signs differ result is negative.

3. Division

• Division by a nonzero rational is multiplying by its reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c), provided c ≠ 0.
• Simplify by canceling common factors before multiplying.

4. Rules for signs

• + ÷ + = +, − ÷ − = +, + ÷ − = −, − ÷ + = − (same for multiplication).
• Addition/subtraction: think of movement on the number line — adding a negative moves left, adding a positive moves right.

5. Useful properties

• Closure: Rational numbers are closed under +, −, ×, ÷ (except division by 0).
• Commutative: a+b = b+a, a×b = b×a.
• Associative: (a+b)+c = a+(b+c), (a×b)×c = a×(b×c).
• Distributive: a×(b+c) = a×b + a×c.
• Identity elements: additive identity = 0, multiplicative identity = 1.
• Inverses: additive inverse of a/b is −a/b; multiplicative inverse of a/b (a ≠ 0) is b/a.

6. Converting mixed numbers and improper fractions

• Mixed to improper: m n/d = (m×d + n) / d (keep sign of whole number).
• Improper to mixed: divide numerator by denominator to get whole part and remainder.

7. Practical tips

• Always reduce the final answer to lowest terms.
• When adding/subtracting many fractions, convert all to a common denominator using LCM to avoid large numbers.
• Cancel early in multiplication/division to keep numbers small.

Definition. A rational number is any number that can be written as p/q where p and q are integers and q ≠ 0. Examples: 3/4, −5, 0.25 (=1/4), 7/3.

Main properties.

• Closure under addition, subtraction and multiplication: If a = p/q and b = r/s (q,s ≠ 0), then (a + b) = (ps + rq)/qs, (a − b) = (ps − rq)/qs, (a·b) = (pr)/(qs). The results are rational.
• Not closed under division by zero: Division a ÷ b is rational only if b ≠ 0. If b = 0, the operation is undefined.
• Commutative laws: a + b = b + a and a·b = b·a for all rationals a, b.
• Associative laws: (a + b) + c = a + (b + c) and (a·b)·c = a·(b·c).
• Distributive law: a·(b + c) = a·b + a·c.
• Identities: Additive identity 0 satisfies a + 0 = a. Multiplicative identity 1 satisfies a·1 = a.
• Inverses: Every rational a = p/q has an additive inverse −a = −p/q so a + (−a) = 0. If a ≠ 0 then it has a multiplicative inverse 1/a = q/p so a·(1/a) = 1.
• Density: Between any two distinct rationals there is always another rational. For example the midpoint (a + b)/2 is rational and lies between a and b.
• Decimal representation: Every rational number has a decimal expansion that either terminates (e.g. 1/4 = 0.25) or repeats periodically (e.g. 2/3 = 0.666...).
• Representation on number line: Rational numbers correspond to points on the number line at fractional positions between integers; they can be located exactly by constructing the appropriate fraction of a unit segment.

How to use these properties: They allow safe manipulation of fractions (change to a common denominator for addition/subtraction, multiply numerators and denominators for product, invert divisor for division) and reasoning about order, closeness (density) and decimal forms.

Definition: A rational number is a number that can be expressed as p/q, where p and q are integers and q ≠ 0. A rational number between two given rational numbers a and b is any rational number r such that a < r < b (or b < r < a).

Key fact (Density): Between any two distinct rational numbers there are infinitely many rational numbers. In particular, the arithmetic mean (a + b)/2 is one rational number between a and b. Repeating the process or using a general formula produces infinitely many.

General method to get n rational numbers between a and b: For any natural number n, the n numbers ((n+1−k)·a + k·b)/(n+1) for k = 1, 2, ..., n are n rational numbers lying strictly between a and b. These give n equally spaced rationals between a and b.

Alternate practical method (common denominator / scaling): If a = p/q and b = r/s, first write both with a common denominator Q = q·s: a = (p·s)/Q, b = (r·q)/Q. If there are integers between p·s and r·q, each corresponding fraction with denominator Q is a rational number between a and b. If not enough integers lie between them, multiply numerator and denominator of both fractions by (n+1) (or a larger integer) to create room to pick integers in between.

Why this works (sketch): The formula ((n+1−k)a + k b)/(n+1) is a convex combination of a and b, hence lies between them. Varying k gives distinct intermediate points; letting n grow or repeating midpoint division gives infinitely many.

Important notes: These methods work for any two rational numbers (positive, negative or of opposite signs). The same idea shows rationals are dense in the real line: between any two real numbers there is a rational number.

What it means: Comparison and ordering of rational numbers means deciding which of two (or more) rational numbers is larger/smaller and arranging a set of rational numbers in ascending or descending order.

Key ideas and rules:

• Rational numbers include positive fractions, negative fractions, and integers (all of which can be written as fractions a/b with b ≠ 0).
• If two numbers have different signs, any positive number is always greater than any negative number. Example: 3/4 > −2/5.
• For two positive rational numbers with the same denominator, the one with the larger numerator is larger. Example: 5/7 > 3/7.
• For two negative rational numbers with the same denominator, the one with the larger numerator is actually smaller (more negative). Example: −5/7 < −3/7.
• If denominators differ, convert to a common denominator (usually use LCM) or convert to decimals, then compare.
• Cross-multiplication rule (for positive denominators): a/b > c/d ⇔ ad > bc. This also works when considering sign carefully (multiply both sides by bd if bd > 0).
• On the number line, numbers increase from left to right. To order numbers, place them on the number line and read left (smaller) to right (larger).

Step-by-step method to compare two rational numbers a/b and c/d:

1. Check signs: if one is positive and the other negative, the positive one is larger.
2. If both have the same sign, either:
   • Convert both to equivalent fractions with the same positive denominator (use LCM of |b| and |d|), then compare numerators; or
   • Use cross-multiplication: compare ad and bc (careful with signs); or
   • Convert to decimal form and compare decimals.
3. To order a list of rational numbers, compare pairwise or convert all to a common form (common denominator or decimals) and sort.

Important intuition: For positives larger numerator/ smaller denominator tends to increase value. For negatives, the number that is farther from zero (larger absolute value) is smaller.

Word problems and applications of rational numbers ask you to model real situations using fractions, negative numbers and ratios, then perform arithmetic to find an answer. The key is translation: read the situation carefully, assign signed rational numbers, choose representations (fraction, mixed number or decimal), perform correct operations, and interpret the result in context.

General strategy: (1) Identify what is given and what is asked. (2) Represent quantities as rational numbers with correct signs (e.g., temperatures below zero are negative). (3) If needed, convert mixed numbers to improper fractions or decimals. (4) For addition/subtraction use a common denominator; for multiplication multiply numerators and denominators; for division multiply by the reciprocal. (5) Simplify and convert to the required form. (6) Check the result in the context of the problem (units, sign, reasonableness).

Common contexts: temperature changes, bank balances (debits/credits), elevations above/below sea level, profit/loss, mixing solutions or ingredients, portions of tasks completed, speeds and distances when fractions occur. Visual aids (number line, bar model, pie or area models) make comparisons and operations easier to understand.