Big picture
A temperature below zero and half of a quantity require numbers beyond positive counting numbers. Rational numbers include integers and fractions that can be expressed as p/q, where p and q are integers and q is nonzero. They provide one system for handling positive and negative fractional quantities.
The central skills are equivalence, order and arithmetic. Keep signs attached to the whole value, use common denominators for addition and comparison, and simplify without changing the value. A number line gives an independent way to check whether a result is positive, negative or between two stated bounds.
Learning goals
Define rational numbers and write equivalent representations.
Reduce a number to standard form with a positive denominator.
Locate and compare positive and negative rational numbers.
Perform all four arithmetic operations.
Find rational numbers between given values and verify results.
Core ideas
1. Meaning and examples
A rational number has the form p/q with integer p and nonzero integer q. Examples include 3/5, −7/4, 0/9 and 12/1. Every integer is rational because it can be written with denominator 1. Zero is rational because 0/1 is valid.
A denominator of zero is not allowed: 5/0 does not represent a rational number. The fraction 0/5 equals zero, but reversing its numerator and denominator produces an undefined expression. A fraction often introduces a part of a whole, while a rational number also includes negative fractions and values greater than one. For example, −3/2 can represent a signed quantity and lies to the left of zero.
2. Signs and equivalent forms
Multiplying or dividing numerator and denominator by the same nonzero integer gives an equivalent rational number. Thus 2/3 = 4/6 = 10/15. Multiplying only the numerator changes the value, so 2/3 is not generally equal to 4/3.
A single negative sign can be placed in the numerator, denominator or before the fraction: −3/5 = 3/(−5) = −(3/5). Two negative signs cancel: (−3)/(−5) = 3/5. Normalising the denominator to positive makes comparisons and later operations clearer. For instance, 8/(−12) becomes −8/12, then simplifies to −2/3.
3. Standard form
A rational number is in standard form when its denominator is positive and numerator and denominator have no common factor other than 1. Divide both by their highest common factor and adjust signs if necessary.
For −42/56, HCF(42, 56) = 14, so −42/56 = −3/4. For 18/(−30), first write −18/30, then divide by 6 to obtain −3/5. Standard form concerns representation, not whether the number is positive or proper. The rational number 7/3 is already standard even though its numerator exceeds its denominator. Zero can be represented canonically as 0/1.
4. Number-line representation
Divide the interval between consecutive integers into equal parts determined by a suitable denominator. To mark 3/4, divide 0 to 1 into four equal parts and select the third. To mark −3/4, move three such parts left from zero.
For −5/3, write −1⅔ or identify it between −2 and −1. Divide that interval into thirds and place the point one-third to the right of −2. Positive rational numbers lie right of zero and negative ones left. Equal rational numbers such as −1/2 and −2/4 occupy the same point, showing that different written forms need not be different numbers.
5. Comparison
With equal positive denominators, compare numerators. Since −7 < −3, −7/8 < −3/8. A negative number with larger magnitude lies farther left and is therefore smaller. For unlike denominators, use equivalent fractions with a positive common denominator.
Compare −2/3 and −3/5. The common denominator 15 gives −10/15 and −9/15, so −2/3 < −3/5. Cross-multiplication is another method when denominator signs are controlled, but a common-denominator solution makes the order transparent. Every positive rational number exceeds every negative one, and zero lies between them. Do not compare only the written numerators when denominators differ.
6. Addition
For the same denominator, add numerators: a/q + b/q = (a + b)/q. For different denominators, rewrite using a common denominator. A positive and negative value may partially cancel.
For −3/4 + 5/6, use denominator 12: −9/12 + 10/12 = 1/12. For −2/7 + (−3/7), add −2 and −3 to get −5/7. Never add denominators: 1/2 + 1/3 = 3/6 + 2/6 = 5/6, not 2/5. On a number line, addition corresponds to signed displacement from the first number.
7. Subtraction and additive inverses
The additive inverse of a is −a because their sum is zero. Subtraction is addition of the additive inverse: a − b = a + (−b). For negative b, this explains the familiar double-negative change.
Compute 2/5 − (−3/10). Rewrite as 2/5 + 3/10 = 4/10 + 3/10 = 7/10. For −1/3 − 1/6, use −2/6 − 1/6 = −1/2. Subtraction is not commutative: 1/2 − 1/3 = 1/6, while 1/3 − 1/2 = −1/6. Preserve the original order before simplifying signs.
8. Multiplication and reciprocals
Multiply numerators and denominators: (a/b)(c/d) = ac/bd. Equal signs give a positive product and unlike signs a negative product. For (−4/9) × (15/8), cancel 4 with 8 and 15 with 9, giving −(1 × 5)/(3 × 2) = −5/6.
The reciprocal of a nonzero rational a/b is b/a because their product is 1. The reciprocal of −3/7 is −7/3, not +7/3. Zero has no reciprocal because division by zero is undefined. Multiplication by 1 leaves a value unchanged, while multiplication by 0 gives zero. Cancellation is valid across factors, not across terms connected by addition.
9. Division
Divide by a nonzero rational number by multiplying by its reciprocal: (a/b) ÷ (c/d) = (a/b)(d/c), requiring c ≠ 0. For (−5/6) ÷ (10/9), compute (−5/6) × (9/10). Cancelling gives −3/4.
Division by a fraction smaller than 1 can increase magnitude. For 3/4 ÷ 1/2, the result is 3/2, answering how many halves fit in three-quarters. Zero divided by a nonzero rational is zero; a nonzero value divided by zero is undefined. The statement “division always makes a number smaller” is therefore false, and sign and divisor size must be considered.
10. Numbers between and practical checks
There are infinitely many rational numbers between two distinct rational numbers. Their average, (a + b)/2, lies between them. Between 1/3 and 1/2, average = (2/6 + 3/6)/2 = 5/12. More examples follow by expressing endpoints as 8/24 and 12/24, giving 9/24, 10/24 and 11/24 between them.
For signed change, a temperature rising from −3/2°C by 5/4°C becomes −6/4 + 5/4 = −1/4°C. It is warmer than before but still below zero. A ribbon of length 3/2 m cut into pieces of length 1/4 m gives (3/2) ÷ (1/4) = 6 pieces. Check each answer in context: warming need not cross zero, and a piece count should be a nonnegative whole number in this exact-cutting example.
A reliable approach to questions
Make denominators positive and simplify signs.
Use common denominators for comparison, addition and subtraction.
Use multiplication and reciprocal rules only for the matching operations.
Cancel common factors before large products.
Check sign, magnitude and position on a number line or in the original context.
Connections that matter
Integers supplies sign rules and ordering below zero.
Fractions supplies equivalence, common denominators and division meaning.
Algebra uses rational arithmetic when solving equations and proportions.
Common mistakes
Allowing a zero denominator creates an undefined expression.
Adding denominators in fraction addition changes the calculation.
Dropping the negative sign from a reciprocal changes the value.
Cancelling terms across addition misuses factor cancellation.
Revision checklist
I can identify rational numbers and exclude zero denominators.
I can write standard form and equivalent fractions.
I can compare and locate signed rational numbers.
I can perform all four operations with checks.
I can find several rational numbers between two given values.