Integers: Definition, Properties, and Examples

What Is Integers?

When asking what is integers, we are referring to a foundational set of numbers used in mathematics. The integers definition states that they form a complete set containing positive counting numbers, their negative counterparts, and zero.

This mathematical set is widely denoted by the letter $Z$ (from the German word Zahlen, meaning numbers) and can be written in set notation:

$Z = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$

The set of integers is an extension of the natural numbers, which only include positive counting numbers. By adding zero and the exact opposites of the natural numbers, we create the complete integer family.

Key Ideas and Vocabulary

Understanding integers on a number line requires familiarity with positive and negative numbers. Positive integers are greater than zero, while negative integers are less than zero. Zero itself is neutral, being neither positive nor negative.

Every integer has an additive inverse, which is its opposite on the number line. When you add an integer and its additive inverse, the result is always zero.

The distance an integer is from zero is its absolute value. Since distance cannot be negative, the absolute value is always positive or zero.

Like other number systems, integers can be classified as even and odd numbers. Even integers are evenly divisible by $2$, while odd integers leave a remainder of $1$ or $-1$.

Visual Explanation

A number line is the most effective way to visualize how integers are arranged and structured.

Moving to the right of zero gives increasingly larger positive values, while moving to the left gives increasingly smaller negative values.

We can also visualize how integer sets relate to other mathematical number sets. Integers include all whole numbers but expand that group by including exact negative values.

Worked Examples

Looking at specific integers examples helps reinforce the definition and properties of this number set.

Example 1: Classifying values as integers

Question: Which of the values $-15$, $4.5$, and $\dfrac{20}{5}$ are integers?

Method:

1. Evaluate $-15$: It is a negative whole number, so it is an integer.
   
2. Evaluate $4.5$: It contains a decimal part, so it is not an integer.
   
3. Evaluate $\dfrac{20}{5}$: Simplify the fraction first. $\dfrac{20}{5} = 4$.
   
4. Check the simplified value: $4$ is a positive whole number, so it is an integer.

Answer: The values $-15$ and $\dfrac{20}{5}$ are integers.

Check: Verify that neither $-15$ nor $4$ have fractional or decimal parts.

Example 2: Finding distance using absolute value

Question: What is the distance between $-7$ and $0$ on a number line?

Method:

1. Identify the concept: Distance on a number line is represented by absolute value.
   
2. Write the absolute value expression for the integer: $|-7|$.
   
3. Evaluate the expression: The distance from $-7$ to $0$ is $7$ units.

Answer: The distance is $7$.

Check: Distance is always a positive integer or zero. Since $7$ is positive, the result makes sense.

Example 3: Applying the additive inverse

Question: What is the result of $-18 + 18$?

Method:

1. Identify the relationship between the two integers.
   
2. Notice that $-18$ is a negative integer and $18$ is its positive opposite.
   
3. Recall the additive inverse property: Adding any integer to its exact opposite results in zero.

Answer: The result is $0$.

Check: Moving $18$ units to the right from $-18$ on a number line lands exactly on $0$.

Common Mistakes and Non-Examples

One of the most frequent errors is assuming that all negative numbers are integers. A value like $-3.2$ is negative, but it has a decimal component, meaning it is not a whole unit. Therefore, it is not an integer.

Another mistake is forgetting that zero is an integer. Because it does not represent a positive or negative quantity, some learners exclude it. However, zero is a valid integer that marks the midpoint between positive and negative values.

Finally, do not rely purely on how a number is written. Fractions such as $\dfrac{1}{2}$ are not integers, but a fraction like $\dfrac{8}{4}$ simplifies to $2$, which is an integer. Always simplify a value before deciding.

Real-World Connections

Integers are useful whenever we need to measure quantities that can go above or below a certain baseline.

Temperatures are commonly expressed as integers. A temperature of $22^\circ\text{C}$ is a positive integer indicating warmth, while $-5^\circ\text{C}$ is a negative integer indicating it is freezing.

In banking and finance, integers track changes in account balances. Depositing money into an account is represented by a positive integer, while withdrawing money is tracked as a negative integer.

Elevation also relies on integers. The height of a mountain is a positive integer measured above sea level, while the depth of a submarine is represented as a negative integer below sea level.