Deriving the Addition and Subtraction Rules for Fractions from First Principles

Fractions | Algebra | First Principles

The familiar rules for adding and subtracting fractions can be derived directly from the properties of real numbers. The key idea is that multiplying a number by 1 does not change its value. By writing 1 in a suitable fractional form, both fractions can be expressed with the same denominator.

Assumptions

Let a, b, c, d ∈ ℝ, with b ≠ 0 and d ≠ 0.

Addition of Two Fractions

Begin with two fractions whose denominators are not necessarily equal:

a b + c d
= a b · 1 + c d · 1
= a b · d d + c d · b b
= ad bd + cb db
= ad bd + bc bd
= ad + bc bd
Why is this valid? Since d ≠ 0 and b ≠ 0, d d = 1 and b b = 1. Therefore multiplying the first fraction by d/d and the second fraction by b/b does not change either value.

Why the Denominators Become Equal

After multiplying, the denominators initially appear as bd and db. Since multiplication in ℝ is commutative,

db = bd
cb = bc

Both fractions can therefore be written with the common denominator bd, allowing their numerators to be added directly.

Subtraction of Two Fractions

Exactly the same reasoning applies when the fractions are subtracted.

a b − c d
= a b · 1 − c d · 1
= a b · d d − c d · b b
= ad bd − cb db
= ad bd − bc bd
= ad − bc bd

Combining Both Results

The addition and subtraction derivations have exactly the same structure. The only difference is the operation performed between the two numerators.

General Fraction Rule
a b ± c d = ad ± bc bd
a, b, c, d ∈ ℝ, with b ≠ 0 and d ≠ 0

What the Derivation Shows

Finding a common denominator is not an arbitrary procedure. It is a consequence of multiplying each fraction by a suitable form of 1.

The fractions themselves do not change. Their representations change so that both denominators become equal. Once that has been achieved, the numerators can be combined while the common denominator remains fixed.

Underlying principle: multiplying by 1 preserves a number, while the commutative property of multiplication allows bd and db, and bc and cb, to be written in the same order.

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