What Is An Integer? — Definition & Examples

Integer Definition

An integer is a whole number from the set of negative, non-negative, and positive numbers. To be an integer, a number cannot be a decimal or a fraction. Here is a list of integers:

• $130$
• $-9$
• $0$
• $25$
• $-\mathrm{7,685}$

Integers

Integers are all negative and positive whole numbers, and $0$. Integers or integer values are part of various numbering systems.

Numbering systems are ways of counting and categorizing real and imaginary objects. Integers are one set of numbers or numbering system you use every day.

Common numbering systems you may encounter include all these:

• Real numbers
• Natural numbers
• Integers
• Imaginary numbers
• Rational numbers
• Irrational numbers
• Complex numbers

We usually write our numbers using Arabic numerals: $0, 1, 2, 3$. But we can also represent them using Roman numerals (except for $0$): $I, II, III$.

Set Of Integers

Mathematicians show sets of numbers using these brackets {}, and ellipsis … to show that the numbers continue without end. So we could show a set of integers like this:

$\left\{... -4, -3, -2, -1, 0, 1, 2, 3, 4 ...\right\}$

We understand that the negative numbers keep going, and so do the positive whole numbers.

Here are two different sets of numbers:

1. What does this set mean? $\left\{1, 2, 3, 4, 5 ...\right\}$

Answer – That is the set of all positive integers.

2. How about this one? $\left\{0, 2, 4, 6, 8, 10 ...\right\}$

Answer – That is the set of all non-negative, even integers.

Integers That Are Not Whole Numbers

Negative integers are not whole numbers. Whole numbers are all counting numbers and $0$. A set of integers that are not whole numbers looks like this:

$\left\{-100, -86, -7, -13, -23, -744\right\}$

A negative number that is not a decimal or fraction is an integer, but not a whole number.

Integer Examples

Integers are positive whole numbers and their additive inverse, any non-negative whole number, and the number zero by itself. Here are examples of integers:

• Negative Integers: $-1, -2, -3, -4, -5$ and so on, without end
• Non-negative Integers: $0$ and all positive whole numbers, like $6, 7, 8, 9$, and so on
• Positive Integers: $1, 2, 3, 4, 5$, without end ...
• Zero: $0$ all by itself

Non-Integers Examples

Non-integers are any number that is a decimal, fraction, or mixed unit. These are all not integers:

• Decimals: $3.14; 0.428571428571; 1.414213562373095$
• Fractions: ½; ¾; ⅞; 1½; 9/3; 3/7
• Mixed units: 7’ - 3½”

Where Do We Use Integers?

Integers pop up in most things you count each day:

• Temperature: $45 °C, -20 °C, 76 °F, 273.16 K$
• Grades: $65\%, 100\%, 50\%$
• Teams: $11$ football players; $9$ baseball players; $four$ four-square players; $six$ members of the Chess Club
• Altitude: Commercial planes fly at $\mathrm{35,000}$ feet above sea level, and higher
• Coins: You have $three$ quarters and $two$ dimes, so you have $95$ cents

Whenever you are counting real objects such as books, pencils, shoes, hats, friends, or the number of hairs on your head, you are counting with integers.

Also, every time you round a number to a whole value, you change it from a decimal, fraction or mixed unit to an integer:

• $5\text{'} 6"$ becomes the integer $6\text{'}$
• $3\frac{7}{8}$ becomes the integer $4$
• $98.6 °F$ becomes the integer $99 °F$
• $\$1.97$ becomes the integer $\$2$
• $364.75$ miles becomes the integer $365$ miles

Characteristics Of Integers

Testing to see if a number is an integer is as easy as asking two questions:

1. Is it a whole number? – Integer!
2. Is it the number 0? – Integer!
3. Is it negative of a whole number? – Integer!
4. Does it include multiple parts, such as $5\text{'} 6"$ or $\$1.97$ – Not an integer!

Which of the following are integers?:

1. $One$
2. $3.14159265358979$
3. $-15 °C$
4. $\frac{11}{3}$
5. $9\text{'} 6\frac{1}{2}"$
6. $-365$
7. $IV$ (Roman numerals!)
8. $\$17.76$
9. $\frac{5}{16}$

Only $\#1, 3, 6,$ and $8$ are integers. The other numbers are perfectly fine numbers, but they are decimals, fractions, and mixed units.