Integers are the basis of mathematics, as such it is important that you are able to master them before moving on to more complex or advanced math concepts such as additive inverses, multiplicative inverses, arithmetic operations, and real numbers.
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What is an integer?
From the set of negative and positive numbers, including zero, an integer is a number with no decimal or fractional element such as -5, 0, 1, 5, 8, 97, and 3043.
There are two types of integers:
- Positive Integers: If an integer is more than zero, it is considered positive. For example, 1,2,3,4,5…
- Negative Integers: If an integer is less than zero, it is considered negative. For instance, -1, -2, -3, -4, -5…
What about zero?
On the number line, zero is the number that falls exactly in the middle of the positive and negative numbers. Along with the positive natural numbers (1, 2, 3, 4,…) and the negative numbers (…-4,-3,-2,-1), zero is considered an integer.
Zero is unique because it is the only integer that is neither positive nor negative. It is also the only integer that isn’t both prime and composite. It’s an even number because it’s divisible by two and has no remainder. In numerous systems of algebra, zero is the additive identity element, and the digit “0” is employed as a placeholder value in positional notation methods for representing numbers.
What about fractions and decimals?
Integers are whole numbers which means they do not include fractions and decimals. Therefore, if you see 4.5, -9.1, ½, ¾, etc., remember that these are not integers.
What is the difference between whole number and integer?
Whole numbers are a collection of positive and zero numbers that do not contain a decimal point or fractions. It should be noted that while all whole numbers are integers, not all integers are whole numbers.
Integers are made up of negative, positive, and zero digits, whereas whole numbers are made up of solely positive and zero digits. As a result, integers contain whole numbers and are often denoted by the integer symbol (Z).
Integers: -3, -2, -1, 0, 1, 2, 3
Whole Numbers: 0, 1, 2, 3
What are integers used for in everyday life?
Highway speed restrictions, clocks, addresses, thermometers, hockey scores, altitude levels, maps, and money are just a few examples of how integers and numbers are used in everyday life.
What jobs use integers?
- Science
Although some scientists work with decimal numbers, a lot of them work with integers. Many measures that can be accomplished with decimals are usually expressed in whole numbers and integers. For example, 3000 light years, -1000 feet, or 200 meters.
- Sports
Many sports statistics use integers and may even contain negative values for very poor achievements. For example, in football, yards are used to express gains and losses, such as a six-yard run (reported as 6 yards) or a three-yard loss (described as -3 yards). Plus/minus is a statistic in hockey that is measured as a positive or negative integer.
- Computer Programming
Integers are used in computer programming as well. Integer functions are available in several programming languages and number systems to convert numbers to positive or negative whole numbers. Integers can also be used to count the number of loops a program performs or to measure the position of a data point within a set.
Which is the smallest integer?
This is a bit of a tricky question. Many people would say zero because it’s the equivalent of nothing. Integers, however, can venture into the negative, or minus realm, and therefore -1 is smaller than 0. If -1 is smaller, then -2 is even smaller than that… therefore the smallest integer is negative infinity and the largest integer is positive infinity.
What are the properties of an integer?
If we were to visualize a number line of a set of integers, all integers left of zero are called negative integers, and all integers to the right of zero are positive integers. However, there are 5 other properties of integers that you should be aware of.
Closure Property
This property, which involves addition and subtraction, states that the combination of any two integers will always be an integer. For example:
7 – 4 = 3
-3 + 2 = -1
This is the same for multiplication and division. For example:
5 x 8 = 40
-4 x 7 = 28
Associative Property
This property refers to the fact that no matter the grouped order of the integers in an equation, the answer will always remain the same. This is better understood by an example:
(3 + 4) + 2 = 3 + (2 + 4)
The answer for both is 9.
This is the same for multiplication. For example:
(4 x 5) x 3 = 5 x (4 x 3)
Both answers equal 60.
However – this rule does not work for subtraction and division.
Commutative Property
This property explains that numbers can be interchangeable within the equation and still reach the same answer. Like the associative property, this holds true for multiplication but not division or subtraction.
For example:
7 + 3 = 10
3 + 7 = 10
Equally:
2 + 25 + 10 + 5 = 42
10 + 5 + 2 + 25 = 42
Distributive Property
According to this property, multiplying a number by a set of numbers added together is the same as multiplying each number separately.
For example:
3 x (2 + 4) = 18
Or, 3 x (6) = 18
By the distributive property:
3 x (2 + 4) is the same as 3 x 2 + 3 x 4
Identity Property
This property states that any integer added to zero will result in the same number.
5 + 0 = 5
1600 + 0 = 1600
This property also states that if an integer is multiplied by 1, the integer itself will be the answer. If the integer is multiplied by 0, then the result will be zero. Lastly, if the integer is multiplied by -1 then the result is the negative of the integer. Here are some examples:
5 x 1 = 5
19 x 1 = 19
Then:
5 x 0 = 0
19 x 0 = 0
Then:
5 x -1 = -5
19 x -1 = -19
What are some examples of integers?
Example 1
Q: (-12) – 16 + (-22) – (33 – 58) = ?
= (-12) – 16 + -22 – (33 – 58)
= (-12) -16 – 22 – (33 – 58)
= -12 -16 -22 – -25
= -12 -16 -22 + 25
= -25 (Negative Integer)
Example 2
Q: (-10) – (-22) + 33 = ?
= (-10) – (-22) +33
= -10 – -22 + 33
= -10 + 22 + 33
= 45 (Positive Integer)
Example 3
Q: (-29 + 4) – (20 +100) = ?
= (-29 + 4) – (20 + 100)
= -25 – 120
= -145 (Negative Integer)
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