Number

What this area is about

Numbers start as answers to “how many?” and grow each time a question has no answer yet. Whole numbers cannot go below zero, so the integers add negatives; one integer does not always divide another evenly, so the rational numbers add fractions; \displaystyle \sqrt{2} is not a fraction, so the real numbers fill the line; no real number squares to -1, so the complex numbers add i. Along the way come writing numbers in place value and computing with them.

How these fit together

Foundations. Counting and cardinality makes counting words tell how many. Place value writes any whole number or decimal with ten digits. Arithmetic gives the four operations on whole numbers and decimals.

Structure. Divisibility and primes breaks whole numbers into prime factors. Exponents and roots write repeated multiplication and undo it.

Number systems. Integers, rational numbers, irrational and real numbers and complex numbers, each containing the one before.

A path through it

Counting first. Place value and arithmetic grow together; exponents come after place value. Divisibility needs arithmetic, place value and exponents. Integers (after exponents) and rational numbers (after divisibility) feed each other, and both come before the real numbers. Complex numbers need exponents and roots.

Next come algebra, where letters stand for numbers and the rules of arithmetic still hold, measurement and statistics.