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CHAPTER
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Addition and Subtraction
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Let s take a close look at the processes, also called operations, known as addition and subtraction Much of this material will seem like a review of arithmetic to you, but you ll need to know it forward and backward to work with the algebra to come later
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Moving Up and Down
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Adding a number to another, or subtracting a number from another, are sophisticated ways of counting When you do these operations with integers, it s like moving up or down, point-bypoint, on a vertical number line of the sort you saw in the last chapter
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Absolute value Imagine the number reflector from Chap 3 as a flat plane perpendicular to the number line and passing through 0, as shown in Fig 4-1 Every number is a certain distance above or below the number reflector The distance of an integer from the number reflector is called the absolute value of the integer To denote absolute value, you enclose a numeral or expression between vertical lines The absolute value of 2 is written |2|, and the absolute value of 3 is written | 3| If you have any quantity, no matter how complicated, you can always indicate its absolute value by putting vertical lines on either side of the set of symbols that represents it There s no such thing as negative absolute value, because it is an expression of distance without taking direction into account To say that a number has an absolute value of 3 is like saying that your house is 3 miles from your cousin s house on the other side of town That s nonsense! You can talk about direction as well as distance Then negative values are possible When you move or travel over a certain distance in a certain direction, it is called displacement, and it can be positive or negative It can even go off in other directions, such as west, or straight up, or toward the sun, or toward your cousin s house The absolute value of any natural number (or nonnegative integer) is equal to that natural number The absolute value of any negative integer is its image in the number reflector If
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52 Addition and Subtraction
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3 2 |2| = 2 1 0 -1 |-3| = 3 -2 -3
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Figure 4-1 The absolute value of a number is its distance from 0 along
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the number line The direction (up or down, positive or negative) doesn t matter That is why absolute values can never be negative
you have a numeral that represents a negative integer, you can get the numeral representing its absolute value by removing the minus sign
Meet the variables! When you want to talk about how numbers relate to each other but don t want to specify any particular numbers, you can use variables instead For variables representing integers, mathematicians most often use small, italic letters from a through q When you see something like a + b = c, you know you are supposed to add a quantity a to another quantity b to get a third quantity c You don t have to know what the actual numbers are, but only that they are related in a certain way The term variable means that a quantity doesn t have any fixed value; it can vary To add, move upward Now let s get back to displacement We ll go up and down here, because we ve already illustrated the number line in a vertical sense Think of upward distances as positive displacements, and downward distances as negative displacements If we have an integer a and we want to add another integer b to it, we first find the point on the number line representing a Then we move up b units That will get us to the point representing a + b As an example, suppose a = 3 and b = 2 We start at the point for 3 and move up 2 units That gets us to the point for 3 + 2 It happens to be 1, as shown on the left side of Fig 4-2
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