Exact calculation with fractions, surds and π
Number • Structure and calculation
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Key concepts
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Exact arithmetic with fractions
Fractions represent parts of a whole as a ratio of two integers. Common denominators enable addition and subtraction because equal denominators produce direct comparison and combination of numerators. Multiplication and division of fractions follow direct rules: multiply numerators and denominators for multiplication; invert the second fraction and multiply for division. Cancellation before multiplication reduces factors and preserves exactness.
Definition and simplification of surds
A surd is an expression containing an irrational root that cannot be simplified to a rational number, typically a square root such as √2 or √3. Simplification extracts perfect square factors so that √(a × b) = √a × √b when a is a perfect square. Simplifying reduces radicals to the form k√m where m has no square factors. Extraction of perfect squares makes comparison and arithmetic on surds straightforward and prevents hidden factors inside roots.
Operations with surds
Addition and subtraction of surds require like radicals. Only surds with the same root part combine directly because unlike surds represent different irrational values. Multiplication and division use distributive and index rules; multiplication of radicals follows √a × √b = √(ab), which can produce new perfect-square factors to simplify. Rational numbers multiply through to allow simplification and factoring. Expansion and factorising produce equivalent surd expressions that reveal common terms for combination.
Multiples of π
Angles and arc lengths often appear as multiples of π. Representation as kπ, where k is a rational number, preserves exact angular or circular measure. Trigonometric exact values at standard angles and areas or circumferences retain exactness when left in terms of π. Arithmetic with multiples of π follows fraction rules for the coefficient k. Simplification requires common denominators for addition and conversion to simplest fractional coefficient form.
Rationalising denominators
A rationalised denominator contains no irrational factors. Rationalising removes surds from the denominator to present an expression in standard simplified form. For a single-term root, multiply numerator and denominator by the same root; for a binomial with roots, multiply by the conjugate to use the difference of squares and eliminate irrational terms. Rationalisation produces equivalent exact values and often simplifies further arithmetic or comparison between expressions.
Key notes
Important points to keep in mind