3.1 Introduction to Integers

Understanding Integers

Integers are the set of whole numbers and their negative counterparts: $\{..., -3, -2, -1, 0, 1, 2, 3, ...\}$. They show up constantly in real life, from temperatures dropping below zero to bank accounts going into the negative. Getting comfortable with integers now sets you up for everything else in algebra.

Plotting Integers on Number Lines

A number line is a straight line where each point corresponds to a number. Zero sits at the center, positive integers go to the right, and negative integers go to the left.

• Positive integers (1, 2, 3, ...) are to the right of zero
• Negative integers (-1, -2, -3, ...) are to the left of zero
• Zero is neither positive nor negative

The farther an integer is from zero on the number line, the greater its absolute value. For example, -4 is farther from zero than -2, so -4 has a greater absolute value (4 units vs. 2 units).

Ordering Integers

The key rule: whichever integer is farther to the right on the number line is the greater number. This works for all integers, but it trips people up with negatives.

• 4 is greater than 2 because 4 is to the right of 2
• -1 is greater than -3 because -1 is to the right of -3 (closer to zero)
• Zero is greater than every negative integer and less than every positive integer

To order a set of integers from least to greatest, just think about where each one falls on the number line, starting from the left:

A common mistake: thinking -5 is greater than -2 because 5 is bigger than 2. Always picture the number line. -5 is farther left, so it's the smaller number.

Opposites of Integers

The opposite of an integer is the number that's the same distance from zero but on the other side of the number line.

• The opposite of 4 is -4 (both are 4 units from zero)
• The opposite of -7 is 7 (both are 7 units from zero)
• The opposite of 0 is 0

To find an opposite, just change the sign. Positive becomes negative, negative becomes positive. The opposite of a number is also called its additive inverse, because when you add a number and its opposite, you always get zero: $4 + (-4) = 0$.

Absolute Value

Absolute value measures how far an integer is from zero on the number line, regardless of direction. It's written with vertical bars: $|x|$.

• $|5| = 5$ because 5 is 5 units from zero
• $|-5| = 5$ because -5 is also 5 units from zero
• $|0| = 0$

Absolute value can never be negative. It only tells you distance, and distance is always zero or positive.

Writing Integer Expressions from Descriptions

Translating words into math expressions is a skill you'll use throughout algebra. Here are the most common keywords:

Addition ("sum," "plus," "more than," "increased by"):

• "The sum of 5 and 3" → $5 + 3$
• "7 more than a number $x$" → $x + 7$

Subtraction ("difference," "minus," "less than," "decreased by"):

• "The difference between 10 and 6" → $10 - 6$
• "4 less than a number $y$" → $y - 4$

Multiplication ("product," "times," "multiplied by"):

• "The product of 3 and 8" → $3 \times 8$
• "5 times a number $z$" → $5z$

Watch out for "less than" because the order flips. "4 less than $y$" is $y - 4$, not $4 - y$. The number after "less than" comes first in the expression.

Looking Ahead

A few terms worth knowing as you move forward with integers:

• Signed numbers is just another name for positive and negative numbers (including zero)
• Integer operations (adding, subtracting, multiplying, and dividing integers) will be covered in upcoming sections
• Integers are part of a bigger family called real numbers, which also includes fractions and decimals