18. The multiplication problem has two negatives, and 2 is an even number.
4. 144. The multiplication problem has two negatives.
5. – 2. The multiplication problem has five negatives.
6. 60. The multiplication problem has two negatives.
The rules for dividing signed numbers are exactly the same as those for multiplying signed numbers – as far as the sign goes (see “Multiplying Signed Numbers” earlier in this chapter.) The rules do differ though because you have to divide, of course.
Remember: When you divide signed numbers, just count the number of negative signs in the problem – in the numerator, in the denominator, and perhaps in front of the problem. If you have an even number of negative signs, the answer is positive. If you have an odd number of negative signs, the answer is negative.
Examples
Q.
A. There are two negative signs in the problem, which is even, so the answer is positive. The answer is +4.
Q.
A. There are three negative signs in the problem, which is odd, so the answer is negative. The answer is –9.
Practice Questions
1.
2.
3.
4.
5.
6.
Practice Answers
1. 2. The division problem has two negatives.
2. – 8. The division problem has one negative.
3. – 6. Three negatives result in a negative.
4. 30. The division problem has two negatives.
5. – 4. The division problem has five negatives.
6. – 1. The division problem has one negative.
What role does 0 play in the signed-number show? What does 0 do to the signs of the answers? Well, when you’re doing addition or subtraction, what 0 does depends on where it is in the problem. When you multiply or divide, 0 tends to just wipe out the numbers and leave you with nothing.
Here are some general guidelines about 0:
✓ Adding zero: 0 + a is just a. Zero doesn’t change the value of a. (This is also true for a + 0.)
✓ Subtracting zero: 0 – a = –a. Use the rule for subtracting signed numbers: Change the operation from subtraction to addition and change the sign of the second number, giving you 0 + (–a). But changing the order, a – 0 = a. It doesn’t change the value of a to subtract 0 from it.
✓ Multiplying by 0: a × 0 = 0. Twice nothing is nothing; three times nothing is nothing; multiply nothing and you get nothing: Likewise, 0 × a = 0.
✓ Dividing 0 by a number: 0 ÷ a = 0. Take you and your friends: If none of you has anything, dividing that nothing into shares just means that each share has nothing.
Remember: You can’t use 0 as a divisor. Numbers can’t be divided by 0; not even 0 can be divided by 0. The answers just don’t exist.
So, working with 0 isn’t too tricky. You follow normal addition and subtraction rules, and just keep in mind that multiplying and dividing with 0 (0 being divided) leaves you with nothing – literally.
Practice Questions
1. 4 + 0 =
2. 0 – 4 =
3. 4 × 0 =
4.
Practice Answers
1. 4. Adding 0 to a number doesn’t change the number.
2. – 4. Change the problem to 0 + (–4) and add.
3. 0. Multiplying by 0 always gives you 0 as a result.
4. 0. Dividing 0 by a nonzero number always gives you 0.
Chapter 3
Incorporating Algebraic Properties
In This Chapter
Algebra has rules for everything, including a sort of shorthand notation to save time and space. The notation that comes with particular properties cuts down on misinterpretation because it’s very specific and universally known. (I give the guidelines for doing operations like addition, subtraction, multiplication, and division in Chapter 2.) In this chapter, you see the specific rules that apply when you use grouping symbols and rearrange terms. You also find how opposites attract – or not – in the form of inverses and identities.
The most commonly used grouping symbols in algebra are (in order from most to least common):
✓ Parentheses ( )
✓ Brackets [ ]
✓ Braces { }
✓ Fraction lines /
✓ Radicals
✓ Absolute value symbols | |
Here’s what you need to know about grouping symbols: You must compute whatever is inside them (or under or over, in the case of the fraction line) first, before you can use that result to solve the rest of the problem. If what’s inside isn’t or can’t be simplified into one term, then anything outside the grouping symbol that multiplies one of the terms has to multiply them all – that’s the distributive property, which I cover in the next section.
Examples
Q. 16 – (4 + 2) =
A. Add the 4 and 2; then subtract the result from the 16: 16 – (4 + 2) = 16 – 6 = 10.
Q.