Integers – Whole Numbers and Their Negatives

Integers extend whole numbers in a new direction: they include all whole numbers and their negative counterparts, with no fractions or decimals. The set is written ℤ = {…, -3, -2, -1, 0, 1, 2, 3, …} – the symbol ℤ comes from the German word Zahlen, meaning “numbers.”

Negative numbers have a surprisingly long and cautious history. Chinese mathematicians were using negative quantities as early as around 200 BCE, described in the classic text The Nine Chapters on the Mathematical Art, representing positive and negative amounts with different coloured counting rods. Centuries later, the Indian mathematician Brahmagupta (628 CE) gave the first systematic rules for arithmetic with negative numbers, framing them as “debts” opposite to positive “fortunes.” Remarkably, many European mathematicians remained deeply suspicious of negative numbers for another thousand years – some 17th- and 18th-century mathematicians still dismissed them as “absurd” or “fictitious,” even while other mathematicians were already using them productively.

What Are Integers?

Integers are whole numbers that can be positive, negative, or zero, with no fractional or decimal part.

Which of these are integers: -7, 3.5, 0, 100, -0.2?

-7, 0, and 100 are integers. 3.5 and -0.2 both have a fractional part, so they are not.

Finding the Opposite

What is the opposite of 9?

The opposite of a number flips its sign while keeping the same distance from zero: the opposite of 9 is -9.

Real-Life Application

Bank balances: a debt is a negative amount of money.
Elevation: below sea level is a negative height.
Temperature: degrees below zero.

Key Takeaways

Integers are ℤ = {…, -2, -1, 0, 1, 2, …} – whole numbers and their negatives.
Negative numbers were used in China by around 200 BCE, and given systematic rules by Brahmagupta in 628 CE.
Many European mathematicians distrusted negative numbers until surprisingly recently in mathematical history.

Practice: Integers

Is It an Integer?