Practice Makes Perfect Algebra - Carolyn Wheater [2]
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Arithmetic to algebra
In arithmetic, we learn to work with numbers: adding, subtracting, multiplying, and dividing. Algebra builds on that work, extends it, and reverses it. Algebra looks at the properties of numbers and number systems, introduces the use of symbols called variables to stand for numbers that are unknown or changeable, and develops techniques for finding those unknowns.
The real numbers
The real numbers include all the numbers you encounter in arithmetic. The natural, or counting, numbers are the numbers you used as you learned to count: {1, 2, 3, 4, 5, …}. Add the number 0 to the natural numbers and you have the whole numbers: {0, 1, 2, 3, 4, …}. The whole numbers together with their opposites form the integers, the positive and negative whole numbers and 0: {…, −3, −2, −1, 0, 1, 2, 3, …}.
There are many numbers between each pair of adjacent integers, however. Some of these, called rational numbers, are numbers that can be expressed as the ratio of two integers, that is, as a fraction. All integers are rational, since every integer can be written as a fraction by giving it a denominator of 1. Rational numbers have decimal expansions that either terminate or infinitely repeat a pattern .
There are still other numbers that cannot be expressed as the ratio of two integers, called irrational numbers. These include numbers like π and . You may have used decimals to approximate these, but irrational numbers have decimal representations that continue forever and do not repeat. For an exact answer, leave numbers in terms of π or in simplest radical form. When you try to express irrational numbers in decimal form, you’re forced to cut the infinite decimal off, and that means your answer is approximate.
The real numbers include both the rationals and the irrationals. The number line gives a visual representation of the real numbers (see Figure 1.1). Each point on the line corresponds to a real number.
Figure 1.1 The real number line.
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EXERCISE 1.1
For each number given, list the sets of numbers into which the number fits (naturals, wholes, integers, rationals, irrationals, or reals).
1. 17.386
2. −5
4. 0
6. 493
7. −17.5
10. π
For 11-20, plot the numbers on the real number line and label the point with the appropriate letter. Use the following figure for your reference.
11. A = 4.5
13. C = −2.75
14. D = 0.1
18. H = 7.25
20. J = 8.9
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Properties of real numbers
As you learned arithmetic, you also learned certain rules about the way numbers behave that helped you do your work more efficiently. You might not have stopped to put names to those properties, but you knew, for example, that 4 + 5 was the same as 5 + 4, but 5 − 4 did not equal 4 − 5.
The commutative and associative properties are the rules that tell you how you can rearrange the numbers in an arithmetic problem to make the calculation easier. The commutative property tells you when you may change the order, and the associative property tells you when you can regroup. There are commutative and associative properties for addition and for multiplication.
Commutative Property for Addition: a + b = b + a [Example: 5 + 4 = 4 + 5]
Commutative Property for Multiplication: a × b = b × a [Example: 3 × 5 = 5 × 3]
Associative Property for Addition:(a + b) + c = a +(b + c) [Example:(3 + 4) + 5 = 3 +(4 + 5)]
Associative Property for Multiplication:(a × b) × c = a ×(b × c) [Example:(2 × 3) × 4 = 2 ×(3 × 4)]
Two other properties of the real numbers sound obvious, but we’d be lost without them. The identity properties for addition and multiplication say that there is a real number—0 for addition and