 # Less than Zero

We have been working with positive numbers up to this point. If you remember, whole numbers include all of the positive numbers and zero. The set of numbers called integers includes positive and negative numbers. Technically, it is whole numbers and their opposites (2 and -2 or 56 and -56). When you multiply positive and negative numbers, don't worry about the signs until you are done with the problem. It's a lot like decimals in that way of waiting until the end to clean up the answer.

When you are done with a multiplication problem that has positive and negative values, you don't care how many positive values there are. You only need to pay attention to the number of negative factors. Negative factors have the power of negation. They can change the signs of products. After you count them up, you need to figure out if you have an odd or an even number of negative signs. If it is even, you have a positive answer. If you have an odd number of negative factors, your answer is negative.

Question:
Will these problems have positive or negative products?
56 * 47 * (-58) * (-12) = ? (Positive: Even number of negative factors)
(-15) * (-15) * 98 * (-3) = ? (Negative: Odd number of negative factors)

That's kind of it. We'll do an easier example to show you the process, but you really only need to do the multiplication and then count the number of negative factors.

Example:
5 * 6 * (-4) = ?
Steps to Solve:
(1) Write out the problem without the signs.
5 * 6 * 4 = ?
(2) Solve the multiplication problem.
(5 * 6) * 4 = 30 * 4 = 120
(3) Count the number of negative factors in the original problem: 1
(4) Since the number of negative factors is odd, the final answer is negative.
5 * 6 * (-4) = -120

Example (with decimals):
5.2 * (-3.1) * 2 = ?
Steps to Solve:
(1) Write out the problem without the signs.
5.2 * 3.1 * 2 = ?
(2) Solve the problem. (You need to count the decimal places in this example).
5.2 * (3.1 * 2) = 5.2 * 6.2 = 32.24
(3) Count the number of negative factors in the original problem: 1
(4) Since the number of negative factors is odd, the final answer is negative.
5.2 * (-3.1) * 2 = -32.24

If you solved the last example on your own, you did a lot of multiplication steps.
• Multiplied two-digit numbers.
• Came up with a product of two decimals.
• Figured out if the product was positive or negative.

Good work! If you didn't get the right answer, get a piece of scratch paper and try to work it out again. It will be worth the effort so that you understand all of the steps in the process.

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