If you’re just starting out in mathematics, you may have heard of the term “integer” but aren’t quite sure what it means. In this article, we’ll be diving into the concept of integers and exploring their properties and uses in mathematics. Whether you’re a student looking to improve your understanding of math or a beginner trying to get a handle on the basics, this guide is here to help.
What is an Integer
An integer is a whole number that can be positive, negative, or zero. It is a member of the set of numbers denoted by the symbol Z (short for “Zahlen,” the German word for numbers). This set includes all positive integers (1, 2, 3, etc.), negative integers (-1, -2, -3, etc.), and zero (0).
Integers are distinct from fractions and decimals, which are numbers with a fractional component. For example, 0.5 and 1/2 are both fractions, while 3.14 and -2.71 are decimals. Integers, on the other hand, are whole numbers with no fractional component.
Properties of Integers:
- Integers are ordered: This means that integers have a relative value to each other. For example, we can say that the integer 5 is greater than the integer 3, because 5 is further to the right on the number line. Similarly, we can say that -5 is less than -3, because -5 is further to the left on the number line.
- Integers are closed under addition and subtraction: This means that the result of adding or subtracting two integers is always another integer. For example, if we add the integers 5 and 3, we get 8, which is also an integer. Similarly, if we subtract the integers 8 and 5, we get 3, which is also an integer.
- Integers are closed under multiplication and division: This means that the result of multiplying or dividing two integers is always another integer, as long as the divisor is not zero. For example, if we multiply the integers 5 and 3, we get 15, which is an integer. Similarly, if we divide the integers 15 and 5, we get 3, which is also an integer.
Uses of Integers in Mathematics:
Integers have a number of important uses in mathematics, including:
- Counting: Integers are often used to count the number of items in a set. For example, if we have a set of three apples, we can use the integer 3 to represent the number of apples in the set.
- Measuring: Integers are also used to measure quantities, such as length, weight, and time. For example, if we have a weight of 10 kilograms, we can use the integer 10 to represent the weight.
- Ordering: Integers are often used to order items in a list or sequence. For example, if we have a list of students ranked by height, we can use integers to represent their relative positions in the list (e.g. 1st place, 2nd place, 3rd place, etc.).
- Solving equations: Integers are often used to solve mathematical equations, which involve finding the value of an unknown quantity. For example, if we have the equation “x + 3 = 7,” we can use the integer 4 (x = 4) to solve for the value of x.
Operations with Integers:
There are a number of basic operations that can be performed with integers, including:
- Addition: Addition involves combining two or more integers to find their sum. For example, if we add the integers 5 and 3, we get 8.
- Subtraction: Subtraction involves finding the difference between two integers. For example, if we subtract the integer 5 from the integer 8, we get 3.
- Multiplication: Multiplication involves finding the product of two or more integers. For example, if we multiply the integers 5 and 3, we get 15.
- Division: Division involves finding the quotient of two integers. For example, if we divide the integer 15 by the integer 5, we get 3.
- Order of operations: It’s important to remember the order of operations when performing calculations with integers. The order of operations dictates the order in which calculations should be performed, starting with parentheses, followed by exponents, then multiplication and division (performed from left to right), and finally, addition and subtraction (also performed from left to right).
Integers and Absolute Value:
The absolute value of an integer is the distance of the integer from zero on the number line. It is represented by the symbol “| |” and is always a positive number, regardless of whether the integer is positive or negative.
For example, the absolute value of the integer -5 is 5, because -5 is 5 units away from zero on the number line. Similarly, the absolute value of the integer 3 is 3, because 3 is 3 units away from zero on the number line.
The absolute value of an integer is often used to find the distance between two integers. For example, if we want to find the distance between the integers 3 and -5, we can use the absolute value of the difference between the two integers: |3 – (-5)| = |3 + 5| = |8| = 8.
Integers and Inequalities:
Inequalities are statements that compare the values of two or more numbers. They are represented using the symbols “>” (greater than), “>=” (greater than or equal to), “<” (less than), and “<=” (less than or equal to).
For example, the inequality “5 > 3” means that 5 is greater than 3. Similarly, the inequality “3 < 5” means that 3 is less than 5.
Inequalities can also be used to represent sets of integers. For example, the inequality “x > 3” represents the set of all integers that are greater than 3. This set includes the integers 4, 5, 6, and so on.
In this article, we’ve explored the concept of integers and their properties and uses in mathematics. Whether you’re a student looking to improve your understanding of math or a beginner trying to get a handle on the basics, we hope this guide has helped you better understand what integers are and how they’re used in math.