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Monday 24th of June
Rotating a Parabola
Multiplying Fractions
Finding Factors
Miscellaneous Equations
Mixed Numbers and Improper Fractions
Systems of Equations in Two Variables
Literal Numbers
Adding and Subtracting Polynomials
Subtracting Integers
Simplifying Complex Fractions
Decimals and Fractions
Multiplying Integers
Logarithmic Functions
Multiplying Monomials
The Square of a Binomial
Factoring Trinomials
The Pythagorean Theorem
Solving Radical Equations in One Variable
Multiplying Binomials Using the FOIL Method
Imaginary Numbers
Solving Quadratic Equations Using the Quadratic Formula
Solving Quadratic Equations
Order of Operations
Dividing Complex Numbers
The Appearance of a Polynomial Equation
Standard Form of a Line
Positive Integral Divisors
Dividing Fractions
Solving Linear Systems of Equations by Elimination
Multiplying and Dividing Square Roots
Functions and Graphs
Dividing Polynomials
Solving Rational Equations
Use of Parentheses or Brackets (The Distributive Law)
Multiplying and Dividing by Monomials
Solving Quadratic Equations by Graphing
Multiplying Decimals
Use of Parentheses or Brackets (The Distributive Law)
Simplifying Complex Fractions 1
Adding Fractions
Simplifying Complex Fractions
Solutions to Linear Equations in Two Variables
Quadratic Expressions Completing Squares
Dividing Radical Expressions
Rise and Run
Graphing Exponential Functions
Multiplying by a Monomial
The Cartesian Coordinate System
Writing the Terms of a Polynomial in Descending Order
Quadratic Expressions
Solving Inequalities
Solving Rational Inequalities with a Sign Graph
Solving Linear Equations
Solving an Equation with Two Radical Terms
Simplifying Rational Expressions
Intercepts of a Line
Completing the Square
Order of Operations
Factoring Trinomials
Solving Linear Equations
Solving Multi-Step Inequalities
Solving Quadratic Equations Graphically and Algebraically
Collecting Like Terms
Solving Equations with Radicals and Exponents
Percent of Change
Powers of ten (Scientific Notation)
Comparing Integers on a Number Line
Solving Systems of Equations Using Substitution
Factoring Out the Greatest Common Factor
Families of Functions
Monomial Factors
Multiplying and Dividing Complex Numbers
Properties of Exponents
Multiplying Square Roots
Adding or Subtracting Rational Expressions with Different Denominators
Expressions with Variables as Exponents
The Quadratic Formula
Writing a Quadratic with Given Solutions
Simplifying Square Roots
Adding and Subtracting Square Roots
Adding and Subtracting Rational Expressions
Combining Like Radical Terms
Solving Systems of Equations Using Substitution
Dividing Polynomials
Graphing Functions
Product of a Sum and a Difference
Solving First Degree Inequalities
Solving Equations with Radicals and Exponents
Roots and Powers
Multiplying Numbers
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Multiplying Integers

Objective Learn how to multiply both positive and negative integers.

We will be studying multiplication in this lesson. Multiplication may be the most confusing and nonintuitive of the operations when applied to negative integers. We will try to demonstrate how the idea behind multiplication is applied to examples.

Multiplying with Negative Integers

Take a look at the following rules for multiplying negative integers. Have the definition of opposites in mind.

Key Idea

1. When we multiply a positive integer and a negative integer together, the result is the opposite of the number we would get if both integers had been positive.

For example, 2 × ( -3) = -(2 × 3) = -6 and - 5 × 4 = -(5 × 4) = -20.

2. When we multiply two negative integers, the result is the positive integer we would get if both integers had been positive. For example, -3 × (-4) = 3 × 4 = 12.

In summary, (1) multiplying a positive integer times a negative integer gives a negative integer, and (2) multiplying two negative integers gives a positive integer.

Try to answer the following questions in order to make sure you understood the previous properties.

• Will -3 × (-7) be positive or negative?


• Will 457 × ( -325) be positive or negative?


The rules for multiplying two integers can also be remembered using the phrase same signs, positive product; different signs, negative product.


Why Are These Rules True?

Multiplying any integer (positive or negative) by a positive integer is the same as adding “copies” of that integer, where the number of copies is equal to the positive integer. Multiplication by a positive integer can be viewed as repeated addition. That is, 5 × 2 is the same as 2 + 2 + 2 + 2 + 2. Then, 5 × (-2) is then equal to (-2) + (-2) + (-2) + (-2) + (-2). The following example uses this thinking.



What is 5 × (-3)?


The product 5 × (-3) can be found by adding 5 copies of the integer -3. So,

5 × (-3) = (-3) + (-3) + (-3) + (-3) + (-3) = -15.

The same answer can be obtained by applying the first rule in the Key Idea shown earlier.

5 × (-3) = -(5 × 3) = -15

The idea that the product of two negative integers is positive is many times harder to understand.

Why is -5 × (-3) = 15?

You do not need a formal proof of this fact. Simply notice that the Example just showed that 5 × (-3) = -15. It is reasonable for the products -5 × (-3) and 5 × (-3) to be opposite of each other since they cannot be the same. So -5 × (-3) should be the opposite of -15 or 15.

Try to give reasons for why the following fact is true.

Given any number x , -1 · x is the opposite of x. By the rules for multiplying negative integers, if x is positive, then -1 · x is negative and its value is the opposite of the value of 1 · x = x . Similarly, if x is negative, then -1 · x is positive and its value is the opposite of the value of 1 · x .

The only way you will fully understand the concept of multiplication of negative integers is if you become skilled at the calculations.

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