Mathematics.

number systems

Integers

Foundations20 minDifficulty1 out of 10

You should know: natural numbers

Overview

The integers extend the natural numbers by adding zero and the negative whole numbers, forming the set ℤ = {..., -2, -1, 0, 1, 2, ...}. Unlike the natural numbers, the integers are closed under subtraction — for any two integers a and b, a - b is always an integer. This closure is precisely what motivated their invention: solving equations like x + 5 = 3 requires a number system beyond the naturals.

Intuition

If the natural numbers are steps forward from a starting line, integers add steps backward and standing still. Owing someone 5 dollars is naturally represented as -5 — a quantity smaller than owing nothing (0), which is smaller still than having 5.

Formal Definition

Definition

The integers are the set:

Z={…,−3,−2,−1,0,1,2,3,…}\mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}
Definition
Z=N∪{0}∪{−n:n∈N}\mathbb{Z} = \mathbb{N} \cup \{0\} \cup \{-n : n \in \mathbb{N}\}

Built from the natural numbers, zero, and their negations

Notation

NotationMeaning
Z\mathbb{Z}The set of all integers (from German 'Zahlen', numbers)

Properties

Closure under addition, subtraction, multiplication

a,b∈Z  ⟹  a+b, a−b, ab∈Za, b \in \mathbb{Z} \implies a+b,\ a-b,\ ab \in \mathbb{Z}

Not closed under division

a,b∈Z̸  ⟹  ab∈Za, b \in \mathbb{Z} \not\implies \frac{a}{b} \in \mathbb{Z}

Example: 3 and 2 are integers, but 3/2 is not.

Well-ordering (bounded below)

Every non-empty subset of Z bounded below has a least element\text{Every non-empty subset of } \mathbb{Z} \text{ bounded below has a least element}

Additive inverses exist

∀a∈Z,∃(−a)∈Z such that a+(−a)=0\forall a \in \mathbb{Z}, \exists (-a) \in \mathbb{Z} \text{ such that } a + (-a) = 0

Applications

Integer arithmetic underlies indexing, loop counters, and fixed-width integer types in every programming language.

Worked Examples

  1. Moving 5 units right from -8 on the number line.

    −8+5=−3-8 + 5 = -3

Answer: -3

Practice Problems

Difficulty 2/10

Compute -12 - (-7).

Common Mistakes

Common Mistake

Believing integer division always produces an integer.

ℤ is closed under +, -, × but NOT ÷. 7 ÷ 2 = 3.5 is not an integer — this gap is exactly what motivates the rational numbers.

Historical Background

Negative numbers were used practically in ancient China (counting rods, c. 200 BCE) and India for representing debts, but were treated with suspicion in the West for centuries — European mathematicians as late as the 17th–18th centuries called them 'absurd' or 'fictitious' numbers. Full acceptance came only once negative numbers were given a rigorous algebraic foundation in the 19th century.

  1. c. 200 BCE

    Chinese mathematicians use red/black counting rods to represent positive and negative quantities

  2. 628 CE

    Brahmagupta gives explicit rules for arithmetic with negative numbers (debts)

    Brahmagupta

  3. 19th century

    Negative numbers are given a rigorous algebraic foundation as equivalence classes of natural number pairs

Summary

  • ℤ = {..., -2, -1, 0, 1, 2, ...} extends ℕ with zero and negative whole numbers.
  • Integers are closed under addition, subtraction, and multiplication, but NOT division.
  • Every integer has an additive inverse, unlike the natural numbers.
  • Historically resisted in the West for centuries despite practical use in China and India.

References

  1. WebsiteWikipedia — Integer