Operations with Decimals

Adding and subtracting: line up the decimal points vertically, fill in zeros so both numbers have the same length, then add as usual.

Multiplying: ignore the points entirely, multiply as whole numbers, then count how many digits came after the points in total and put that many in the answer. 0.3 × 0.02 = 6, with three decimal places, so 0.006.

Dividing: move the divisor's point to make it whole, move the dividend's point the same number of places, then divide normally. 4.5 ÷ 0.5 becomes 45 ÷ 5 = 9.

Each rule is powers of ten bookkeeping. Writing 0.3 = 3×10⁻¹ and 0.02 = 2×10⁻², the product is 6×10⁻³ — the exponents add, which is precisely the "count the decimal places" rule.

Alignment when adding is the same principle stated differently: you may only add coefficients that sit on the same power of ten, exactly as you may only add like terms in algebra.

Division shifts both numbers by the same power of ten, using a/b = (10ᵏa)/(10ᵏb). The quotient is unchanged because you multiplied by 10ᵏ/10ᵏ = 1. Note that dividing by a number less than 1 produces a larger result, which surprises students until they read it as "how many halves fit in 4.5" rather than as shrinking.