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📈College Algebra

3.1 Introduction to Integers

3 min readLast Updated on June 24, 2024

Integers are the building blocks of math, including whole numbers and their negative counterparts. They're plotted on a number line, with positive numbers to the right of zero and negative numbers to the left. Understanding integers is crucial for grasping more complex mathematical concepts.

Integers help us represent real-world situations involving gains, losses, temperatures, and more. We can compare, order, and find opposites of integers. Absolute value shows an integer's distance from zero, regardless of its sign. These concepts form the foundation for algebra and beyond.

Understanding Integers

Plotting integers on number lines

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  • Integers include whole numbers (positive integers) and their opposites (negative integers)
    • Positive integers (1, 2, 3) are located to the right of zero on a number line
    • Negative integers (-1, -2, -3) are located to the left of zero on a number line
  • Zero (0) is neither positive nor negative and is positioned at the center of the number line
  • The distance an integer is from zero determines its absolute value
    • 5 is further from zero than 3, so 5 has a greater absolute value (5 units) than 3 (3 units)
    • -4 is further from zero than -2, so -4 has a greater absolute value (4 units) than -2 (2 units)

Ordering integers with zero

  • When comparing two integers, the integer positioned further to the right on the number line is greater
    • 4 is greater than 2 because 4 is located to the right of 2 on the number line
    • -1 is greater than -3 because -1 is located to the right of -3 on the number line
  • To order integers from least to greatest, start with the integer furthest to the left on the number line and move rightward
    • Ordered from least to greatest: -5, -2, 0, 3, 7
  • Zero is greater than any negative integer (-1, -2, -3) and less than any positive integer (1, 2, 3)

Opposites of integers

  • The opposite of an integer is the integer that is equidistant from zero on the number line but on the opposite side
    • The opposite of 4 is -4 because they are both 4 units from zero, but on opposite sides
    • The opposite of -7 is 7 because they are both 7 units from zero, but on opposite sides
  • The opposite of zero is zero
  • To find the opposite of an integer, change its sign
    • The opposite of a positive integer (5) is a negative integer (-5)
    • The opposite of a negative integer (-3) is a positive integer (3)
  • The opposite of a number is also known as its additive inverse

Absolute value in expressions

  • The absolute value of an integer is its distance from zero on the number line, regardless of sign
    • The absolute value of 5 is 5 because it is 5 units from zero
    • The absolute value of -5 is also 5 because it is 5 units from zero
  • The absolute value of zero is zero
  • Absolute value is denoted using vertical bars: x|x|
    • 7=7|-7| = 7
    • 4=4|4| = 4

Integer expressions from descriptions

  • Words indicating addition: "sum," "plus," "more than," "increased by"
    • "The sum of 5 and 3" can be written as 5+35 + 3
    • "7 more than a number xx" can be written as x+7x + 7
  • Words indicating subtraction: "difference," "minus," "less than," "decreased by"
    • "The difference between 10 and 6" can be written as 10610 - 6
    • "4 less than a number yy" can be written as y4y - 4
  • Words indicating multiplication: "product," "times," "multiplied by"
    • "The product of 3 and 8" can be written as 3×83 \times 8
    • "5 times a number zz" can be written as 5z5z

Advanced Integer Concepts

  • Signed numbers include both positive and negative integers, as well as zero
  • The coordinate plane is a two-dimensional representation of integer pairs, with a horizontal x-axis and a vertical y-axis
  • Integer operations involve addition, subtraction, multiplication, and division of integers
  • Real numbers encompass integers, rational numbers, and irrational numbers on the number line

Key Terms to Review (21)

Absolute Value: Absolute value is a mathematical concept that represents the distance of a number from zero on the number line, regardless of the number's sign. It is a way to describe the magnitude or size of a number without considering its positive or negative direction.
Additive Inverse: The additive inverse of a number is the opposite value that, when added to the original number, results in a sum of zero. It is a fundamental concept in the context of integers and other number systems.
Coordinate Plane: The coordinate plane is a two-dimensional grid used to represent and analyze the position and relationships of points, lines, and other geometric shapes. It consists of a horizontal x-axis and a vertical y-axis that intersect at a central point known as the origin.
Difference: The difference between two numbers is the amount by which one number exceeds the other. It is the result of subtracting one number from another and represents the magnitude of the separation between the two values.
Equidistant: Equidistant refers to the property of being an equal distance from two or more points or objects. It describes a relationship where the distance between a specific point and multiple other points is the same.
Expressions: Expressions are mathematical phrases that combine numbers, variables, and operations to represent a value or quantity. They are fundamental building blocks used to describe and manipulate numerical relationships in the context of mathematics and algebra.
Greater Than: The term 'greater than' is a mathematical comparison operator that indicates when one value exceeds another. It is a fundamental concept in both the study of integers and the addition and subtraction of fractions with common denominators.
Integer Operations: Integer operations refer to the basic mathematical operations that can be performed on integers, which are whole numbers that can be positive, negative, or zero. These operations form the foundation for working with integers in various mathematical contexts.
Integers: Integers are a set of positive and negative whole numbers, including zero. They are the foundation for many mathematical operations and concepts, and are essential in understanding and working with various topics in pre-algebra.
Least to Greatest: The concept of ordering a set of numbers or values from the smallest or lowest value to the largest or highest value. This arrangement is often used to organize and compare numerical data in a logical and meaningful way.
Less Than: The term 'less than' is a mathematical comparison that indicates a value or quantity is smaller or lower in magnitude than another. This concept is crucial in understanding integers and performing operations with fractions.
Negative Integers: Negative integers are numbers that are less than zero on the number line. They represent quantities or values that are below the starting point of zero, such as debt, losses, or temperatures below freezing.
Number Line: A number line is a visual representation of the number system, where numbers are arranged in a linear fashion along a horizontal or vertical axis. It serves as a fundamental tool in understanding and working with various numerical concepts, including whole numbers, integers, fractions, and rational and irrational numbers.
Opposites: Opposites refer to two things or concepts that are completely different or contrary to each other. They represent the farthest ends of a spectrum or scale, with one being the exact inverse or negation of the other.
Ordering: Ordering refers to the process of arranging or sequencing numbers, quantities, or items in a specific order, such as from smallest to largest or from greatest to least. It is a fundamental concept in mathematics and is particularly relevant in the context of understanding integers.
Positive Integers: Positive integers are the set of all whole numbers greater than zero. They are an important concept in mathematics, particularly in the context of addition and other basic arithmetic operations.
Product: The product is the result of multiplying two or more numbers or quantities together. It represents the combined or cumulative effect of the factors involved in the multiplication operation.
Real Numbers: Real numbers are a comprehensive collection of numerical values that encompass all rational and irrational numbers. They represent the complete set of numbers that can be used to describe and quantify the physical world around us, including measurements, quantities, and mathematical relationships.
Signed Numbers: Signed numbers are a set of numbers that include both positive and negative values, represented by a sign (+ or -) in front of the number. They are used to describe quantities that have direction or can be above or below a reference point.
Sum: The sum is the result of adding two or more numbers or quantities together. It represents the total or combined value of the addends. The sum is a fundamental concept in mathematics that is essential for understanding addition and its applications in various mathematical topics.
Zero: Zero is a fundamental numerical concept that represents the absence of quantity or magnitude. It serves as a starting point and a reference for various mathematical operations and number systems, making it a crucial element in the understanding of whole numbers, integers, and the properties of identity, inverses, and zero.
Absolute Value
See definition

Absolute value is a mathematical concept that represents the distance of a number from zero on the number line, regardless of the number's sign. It is a way to describe the magnitude or size of a number without considering its positive or negative direction.

Term 1 of 21

Key Terms to Review (21)

Absolute Value
See definition

Absolute value is a mathematical concept that represents the distance of a number from zero on the number line, regardless of the number's sign. It is a way to describe the magnitude or size of a number without considering its positive or negative direction.

Term 1 of 21

Absolute Value
See definition

Absolute value is a mathematical concept that represents the distance of a number from zero on the number line, regardless of the number's sign. It is a way to describe the magnitude or size of a number without considering its positive or negative direction.

Term 1 of 21



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© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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