3.1 Introduction to Integers

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: $|x|$
  • $|-7| = 7$
  • $|4| = 4$

Integer expressions from descriptions

• Words indicating addition: "sum," "plus," "more than," "increased by"
  • "The sum of 5 and 3" can be written as $5 + 3$
  • "7 more than a number $x$" can be written as $x + 7$
• Words indicating subtraction: "difference," "minus," "less than," "decreased by"
  • "The difference between 10 and 6" can be written as $10 - 6$
  • "4 less than a number $y$" can be written as $y - 4$
• Words indicating multiplication: "product," "times," "multiplied by"
  • "The product of 3 and 8" can be written as $3 \times 8$
  • "5 times a number $z$" can be written as $5z$

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