Multiplying and Dividing Signed Numbers - Seventh Grade Mathematics
Multiplying and dividing signed numbers extends the logic of repeated equal groups and inverse operations, governed by elegant rules determined by the signs of the factors.
The Logic of Multiplication with Signed Numbers
Multiplication of positive whole numbers represents repeated addition. For example, three times four means adding four three times: four plus four plus four equals twelve. We can use this exact same framework to understand why a positive integer multiplied by a negative integer must yield a negative product.
MULTIPLICATION AS REPEATED ADDITION
3 * (-4) = (-4) + (-4) + (-4) = -12
Three groups of negative four combine to make negative twelve.
Because the commutative property of multiplication guarantees that the order of factors does not change the product, negative four times three must also equal negative twelve. Thus, whenever you multiply two integers with different signs, the product is always negative.
Why Negative Times Negative Equals Positive
The rule that a negative number multiplied by a negative number produces a positive product can be demonstrated through numerical patterns and through algebraic proof using the distributive property.
OBSERVING THE NUMERICAL PATTERN
3 * (-4) = -12
2 * (-4) = -8 (increases by +4)
1 * (-4) = -4 (increases by +4)
0 * (-4) = 0 (increases by +4)
-1 * (-4) = +4 (increases by +4)
-2 * (-4) = +8 (increases by +4)
As the first factor decreases by one in each step, the product increases by four. Continuing this predictable pattern past zero into negative integers forces the products to become positive. We can also prove this algebraically using the distributive property:
Proof that (-1) * (-1) = 1:
We know that: -1 * (1 + (-1)) = -1 * 0 = 0
Distributing: (-1 * 1) + (-1 * -1) = 0
Simplifying: -1 + (-1 * -1) = 0
Add 1 to both sides: (-1 * -1) = 1
This mathematical proof demonstrates that negative times negative equals positive is a logical necessity to keep the fundamental laws of arithmetic consistent.
Summary of Sign Rules for Multiplication and Division
Division is the direct inverse operation of multiplication. Asking what is a divided by b is equivalent to asking what number multiplied by b gives a. Because division mirrors multiplication, the exact same sign relationships apply to both operations.
+-----------------------+-----------------------+--------------------------+
| Mathematical Form | Sign Condition | Sign of Final Result |
+-----------------------+-----------------------+--------------------------+
| (+) * (+) or (+) / (+) | Same signs (both +) | Positive (+) |
| (-) * (-) or (-) / (-) | Same signs (both -) | Positive (+) |
| (+) * (-) or (+) / (-) | Different signs | Negative (-) |
| (-) * (+) or (-) / (+) | Different signs | Negative (-) |
+-----------------------+-----------------------+--------------------------+
When multiplying or dividing two numbers, if the signs are the same, the result is positive. If the signs are different, the result is negative.
Products of Three or More Signed Numbers
When evaluating expressions with multiple factors, the overall sign of the product depends entirely on the total count of negative factors.
Even count of negative factors --> POSITIVE product
Odd count of negative factors --> NEGATIVE product
Example A: (-2) * (-3) * (-4)
[(+6)] * (-4) = -24 (3 negative factors --> Negative)
Example B: (-2) * (-3) * (-4) * (-5)
[(+6)] * [(+20)] = +120 (4 negative factors --> Positive)
Every pair of negative factors multiplies to create a positive value. If all negative factors can be paired up (an even count), the final product is positive. If one negative factor remains unpaired (an odd count), the final product becomes negative.
Division Properties of Zero
Zero plays a unique role in multiplication and division that requires careful attention to avoid logical fallacies.
1. Zero divided by any non-zero number is ZERO:
0 / (-7) = 0 because 0 * (-7) = 0
2. Any number divided by ZERO is UNDEFINED:
(-7) / 0 = undefined because no number times 0 can equal -7
You can divide zero into equal parts, but you can never divide a quantity by zero. Division by zero is undefined throughout mathematics.
Exercises
Work through each problem carefully in your notebook before reading the provided solutions.
Exercise 1. Evaluate the product of negative six and positive nine, and explain the sign rule that justifies your answer.
Exercise 2. Evaluate the quotient of negative seventy-two divided by negative eight, and describe the inverse multiplication relationship that confirms the result.
Exercise 3. Compute the product of the following four factors: negative two times negative three times positive four times negative five.
Exercise 4. A business venture experiences an operating loss of four hundred fifty dollars each month for six consecutive months. Model this scenario with a multiplication expression of signed numbers and determine the total change in the company's financial balance.
Solutions
Solution to Exercise 1. The product of negative six and positive nine is negative fifty-four. This is justified by the rule that multiplying two numbers with different signs always produces a negative result.
Solution to Exercise 2. The quotient of negative seventy-two divided by negative eight is positive nine. In inverse multiplication terms, this division asks what number times negative eight equals negative seventy-two. Since positive nine times negative eight equals negative seventy-two, the quotient must be positive nine. Dividing two numbers with the same sign yields a positive result.
Solution to Exercise 3. Count the number of negative factors in the expression: negative two, negative three, and negative five constitute three negative factors. Because three is an odd number, the final product will be negative. Now multiply the absolute values: two times three equals six; six times four equals twenty-four; twenty-four times five equals one hundred twenty. Applying the negative sign yields a final product of negative one hundred twenty.
Solution to Exercise 4. An operating loss of four hundred fifty dollars per month is represented by negative four hundred fifty. The duration of six months is represented by positive six. The multiplication expression is six times negative four hundred fifty. Multiplying a positive by a negative gives a negative product: six times four hundred fifty equals two thousand seven hundred, so the product is negative two thousand seven hundred. The company's financial balance decreased by two thousand seven hundred dollars.