Integers, Integer Properties and the Role of Zero
Integers are natural numbers (or whole numbers) that stem from the Latin word meaning "intact." In other words, any two integers will be rational numbers. A rational number is a value without fractional part or decimal remainders.
These counting numbers are some of the most common values for both complex arithmetic operations and simple real-life applications using addition and multiplication.
Positive and Negative Numbers
There are two basic types of integers: positive and negative integers. These include any natural number, including zero.
How Are Integers Shown on a Number Line?
Consecutive integers are visually represented on a number line with zero in the middle. Any negative integer value is shown to the left of zero, and positive integers lie on the right.
Positive Integers
On an integer number line, positive integers lie to the right of zero. Each tick mark to the right represents an increase of positive numbers by an absolute value of 1.
Negative Integers
Negative integers lie on the left side of zero. Although these represent negative value increments of negative numbers, the absolute values represent an equal distance from zero as their inverse positive integers.
Is Zero a Positive Integer?
Zero is grouped in with other whole numbers but is not considered a member of either positive or negative numbers. Zero provides the anchor on which both positive or negative integers on a number line are based.
7 Properties of Integers With Examples
The primary properties of integers include the following:
1. Closure Property
The set of integers is closed under addition and multiplication, meaning that the sum or product of any two integers is also an integer.
2. Associative Property
In integer addition and multiplication, the way in which numbers are grouped does not change the result. Here are two examples.
(a + b) + c = a + (b + c)
and
(a x b) x c = a x (b x c)
3. Commutative Property
The order of integers in addition and multiplication does not affect the result. Here are two examples.
a + b = b + a
and
a x b = b x a
4. Distributive Property
Multiplication distributes over addition for integers, meaning:
a x (b + c) = (a x b) + (a x c)
5. Additive Inverse Property
Every integer a has an additive inverse –a such that:
a + (–a) = 0
6. Multiplicative Inverse Property
Every nonzero integer a has a multiplicative inverse 1/a, but since 1/a is typically not an integer, this property mostly applies to rational numbers.
7. Identity Property
The identity element for addition is 0, since a + 0 = a. The identity element for multiplication is 1, since a x 1 = a.
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Now That's Basic
One of the first real-life situations where people encounter integers is counting in the earliest days of education. By calling out repetitive values on a mental integer number line (or even counting on your fingers), your brain begins to identify a pattern called consecutive integers. Building from this mathematical foundation, you can move on to higher-level integer operations, including division and multiplication of integers.
Original article: Integers, Integer Properties and the Role of Zero
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