Whole-number arithmetic

Builds on

Required prerequisites, all the way down: 3

You will learn to

• Multiply and divide multi-digit whole numbers, explaining each step of the algorithms through place value and the distributive law.
• Divide with remainder, writing a = qb + r with 0 ≤ r < b.
• Use the commutative, associative and distributive laws to calculate flexibly, for example 7 × 98 = 7 × 100 − 7 × 2.

Definition

• a × b is the total number of objects in a equal groups of b, or the number of cells in an a-by-b array.
• Division with remainder: for whole numbers a and b with b > 0, there are unique whole numbers q and r with a = qb + r and 0 ≤ r < b.
• The distributive law: a × (b + c) = a × b + a × c.

Theorem

• Multiplication is commutative and associative and distributes over addition.
• The quotient and remainder in division with remainder are unique.
• Division by zero is undefined: for instance, no number q satisfies q × 0 = 5.

Common misconceptions

• Believing multiplication always makes a number bigger (false for 0 and 1, and later for fractions less than 1).
• Allowing a remainder as large as, or larger than, the divisor.
• Treating 'carrying' as a trick rather than as regrouping ten ones into one ten.

Leads to

Optional discoveries

• Rings, ideals and fields The laws that make arithmetic work (commutative, associative, distributive) later become the axioms of a ring.

Learning pathways

• Conventional K–12 progression
• Adult beginner pathway

Further reading

• Liping Ma, Knowing and Teaching Elementary Mathematics (1999)
• Hung-Hsi Wu, Understanding Numbers in Elementary School Mathematics (2011)