$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 3.S: Integers (Summary)

### Key Terms

 absolute value A number's distance from 0 on the number line. integers Counting numbers, their opposites, and zero$$... –3, –2, –1, 0, 1, 2, 3 ...$$ negative number A number less than zero. opposites The number that is the same distance from zero on the number line, but on the opposite side of zero.

### Key Concepts

#### 3.1 - Introduction to Integers

• Opposite Notation
• −a means the opposite of the number a
• The notation −a is read the opposite of a.
• Absolute Value Notation
• The absolute value of a number n is written as |n|.
• |n| ≥ 0 for all numbers.

• Addition of Positive and Negative Integers
 5 + 3 −5 + (−3) both positive, sum positive both negative, sum negative When the signs are the same, the counters would be all the same color, so add them. −5 + 3 5 + (−3) different signs, more negatives different signs, more positives sum negative sum positive When the signs are different, some counters would make neutral pairs; subtract to see how many are left.

#### 3.3 - Subtract Integers

• Subtraction of Integers

#### Table 3.110

 5 – 3 = 2 –5 – (–3) = –2 2 positives 2 negatives When there would be enough counters of the color to take away, subtract. –5 – 3 = 5 – (–3) = 8 5 negatives, want to subtract 3 positives 5 positives, want to subtract 3 negatives When there would not be enough of the counters to take away, add neutral pairs.
• Subtraction Property
• a − b = a + (−b)
• a − (−b) = a + b
• Solve Application Problems
1. Identify what you are asked to find.
2. Write a phrase that gives the information to find it.
3. Translate the phrase to an expression.
4. Simplify the expression.
5. Answer the question with a complete sentence.

#### 3.4 - Multiply and Divide Integers

• Multiplication of Signed Numbers
• To determine the sign of the product of two signed numbers:
Same Signs Product
Two positives Positive
Two negatives Positive
Different Signs Product
Positive • negative Negative
Negative • positive Negative
• Division of Signed Numbers
• To determine the sign of the quotient of two signed numbers:
Same Signs Quotient
Two positives Positive
Two negatives Positive
Different Signs Quotient
Positive & negative Negative
Negative & positive Negative
• Multiplication by −1
• Multiplying a number by −1 gives its opposite: −1a = − a
• Division by −1
• Dividing a number by −1 gives its opposite: a ÷ (−1) = −a

#### 3.5 - Solve Equations Using Integers; The Division Property of Equality

• How to determine whether a number is a solution to an equation.
1. Substitute the number for the variable in the equation.
2. Simplify the expressions on both sides of the equation.
3. Determine whether the resulting equation is true.
• If it is true, the number is a solution.
• If it is not true, the number is not a solution.
• Properties of Equalities
Subtraction Property of Equality Addition Property of Equality Division Property of Equality
For any numbers a, b, c, if a = b then a − c = b − c. For any numbers a, b, c, if a = b then a + c = b + c. For any numbers a, b, c, and c ≠ 0 If a = b, then $$\frac{a}{c} = \frac{b}{c}$$.