Practice Makes Perfect Algebra - Carolyn Wheater [3]
Identity for Addition: a + 0 = a
Identity for Multiplication: a × 1 = a, a ≠ 0
The inverse properties guarantee that whatever number you start with, you can find a number to add to it, or to multiply it by, to get back to the identity.
Inverse for Addition: a + −a = 0 [Example: 4 + −4 = 0]
Inverse for Multiplication: [Example:
Notice that 0 doesn’t have an inverse for multiplication. That’s because of another property you know but don’t often think about. Any number multiplied by 0 equals 0.
Multiplicative Property of Zero: a × 0 = 0
It’s interesting that while multiplying by 0 always gives you 0, there’s no way to get a product of 0 without using 0 as one of your factors.
Zero Product Property: If a × b = 0, then a= 0 or b= 0 or both.
Finally, the distributive property ties together addition and multiplication. The distributive property for multiplication over addition—its full name—says that you can do the problem in two different orders and get the same answer. If you want to multiply 5 × 40 + 8, you can add 40 + 8 = 48 and then multiply 5 × 48, or you can multiply 5 × 40 = 200 and 5 × 8 = 40, and then add 200 + 40. You get 240 either way.
Distributive Property: a(b + c) = a × b + a × c
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EXERCISE 1.2
Identify the property of the real number system that is represented in each example.
1. 7 + 6 + 3 = 7 + 3 + 6
2.(5 × 8) × 2 = 5 ×(8 × 2)
3. 4 + 0 = 4
5. 8(3 + 9) = 8 × 3 + 8 × 9
6. 5x = 0, so x = 0
7.(8 + 3) + 6 = 8 +(3 + 6)
8. 28 × 1 = 28
9. 7 × 4 × 9 = 4 × 7 × 9
10. 193 × 0 = 0
11. 14 +(−14) = 0
12. 3(58) = 3 × 50 + 3 × 8
14.(4 + 1) + 9 = 4 +(1 + 9)
15. 839 +(−839) = 0
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Integers
The integers are the positive and negative whole numbers and 0. On the number line, the negative numbers are a mirror image of the positive numbers; this can be confusing sometimes when you’re thinking about the relative size of numbers. On the positive side, 7 is larger than 4, but on the negative side, −7 is less than −4. It may help to picture the number line and think about “larger” as farther right and “smaller” as farther left.
Expanding your understanding of arithmetic to include the integers is a first big step in algebra. When you first learned to subtract, you would have said you couldn’t subtract 8 from 3, but when you open up your thinking to include negative numbers, you can. The rules for operating with integers apply to all real numbers, so it’s important to learn them well.
Absolute value
The absolute value of a number is its distance from 0 without regard to direction. If a number and its opposite are the same distance from 0, in opposite directions, they have the same absolute values. |4| and |−4| both equal 4, because both 4 and −4 are four units from 0.
Addition
To add integers with the same sign, add the absolute values and keep the sign. Add 4 + 7, both positive numbers, and you get 11, a positive number. Add −5 +(−3), both negative, and you get −8.
To add integers with different signs, subtract the absolute values and take the sign of the integer with the larger absolute value. If you need to add 13 +(−5), think 13 − 5 = 8, then look back and see that the larger-looking number, 13, is positive, so your answer is positive 8. On the other hand, 9 +(−12), is going to turn out negative because |−12| > |9|. You’ll wind up with −3.
Subtraction
Did you notice that none of the properties of the real numbers talked about subtraction? That’s because subtraction is defined as addition of the inverse. To subtract 4, you add −4; to subtract −9, you add 9. When you learned to subtract, to answer questions like 8 − 5 = , what you were really doing was answering + 5 = 8. Every subtraction problem is an addition problem in disguise.
To subtract an integer, add its opposite. Change the sign of the second number(the subtrahend, if you want the mathematical term) and add. The problem 9 −(−7) becomes 9 + 7, whereas −3 −8 becomes