Section A.1 Arithmetic with Negative Numbers
ΒΆpermalinkAdding, subtracting, multiplying, dividing, and raising to powers each have peculiarities when using negative numbers. This section reviews arithmetic with signed (both positive and negative) numbers.
Subsection A.1.1 Signed Numbers
permalinkIs it valid to subtract a large number from a smaller one? It may be hard to imagine what it would mean physically to subtract cars from your garage if you only have car in there in the first place. Nevertheless, mathematics has found a way to give meaning to expressions like using signed numbers.
permalinkIn daily life, the signed numbers we might see most often are temperatures. Most people on Earth use the Celsius scale; if you're not familiar with the Celsius temperature scale, think about these examples:
permalinkFigure A.1.1.2 uses a number line to illustrate these positive and negative numbers. A number line is a useful device for visualizing how numbers relate to each other and combine with each other. Values to the right of are called positive numbers and values to the left of are called negative numbers.
Warning A.1.1.3. Subtraction Sign versus Negative Sign.
Unfortunately, the symbol we use for subtraction looks just like the symbol we use for marking a negative number. It will help to identify when a βminusβ sign means βsubtractβ or means βnegative.β The key is to see if there is a number to its left, not counting anything farther left than an open parenthesis. Here are some examples.
has one negative sign and no subtraction sign.
has no negative signs and one subtraction sign.
has a negative sign and then a subtraction sign.
has two negative signs and no subtraction sign.
Checkpoint A.1.1.4.
Identify βminusβ signs.
Subsection A.1.2 Adding
permalinkAn easy way to think about adding two numbers with the same sign is to simply (at first) ignore the signs, and add the numbers as if they were both positive. Then make sure your result is either positive or negative, depending on what the sign was of the two numbers you started with.
Example A.1.2.5. Add Two Negative Numbers.
If you needed to add and note that both are negative. Maybe you have this expression in front of you:
but that βplus minusβ is awkward, and in this book you are more likely to have this expression:
with extra parentheses. (How many subtraction signs do you see? How many negative signs?)
Since both our terms are negative, we can add and to get and immediately realize that our final result should be negative. So our result is
permalinkThis approach works because adding numbers is like having two people tugging on a rope in one direction or the other, with strength indicated by each number. In Example A.1.2.5 we have two people pulling to the left, one with strength the other with strength Their forces combine to pull left with strength giving us our total of as illustrated in Figure A.1.2.6.
permalinkIf we are adding two numbers that have opposite signs, then the two people tugging the rope are opposing each other. If either of them is using more strength, then the overall effect will be a net pull in that person's direction. And the overall pull on the rope will be the difference of the two strengths. This is illustrated in Figure A.1.2.7.
Example A.1.2.8. Adding One Number of Each Sign.
Here are four examples of addition where one number is positive and the other is negative.
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We have one number of each sign, with sizes and Their difference is But of the two numbers, the negative number dominated. So the result from adding these is
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We have one number of each sign, with sizes and Their difference is But of the two numbers, the positive number dominated. So the result from adding these is
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We have one number of each sign, with sizes and Their difference is But of the two numbers, the negative number dominated. So the result from adding these is
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We have one number of each sign, both with size The opposing forces cancel each other, leaving a result of
Checkpoint A.1.2.9.
Subsection A.1.3 Subtracting
permalinkPerhaps you can handle a subtraction such as where a small positive number is subtracted from a larger number. There are other instances of subtraction that might leave you scratching your head. In such situations, we recommend that you view each subtraction as adding the opposite number.
Original | Adding the Opposite | |
Subtracting a larger positive number: | ||
Subtracting from a negative number: | ||
Subtracting a negative number: |
permalinkThe benefit is that perhaps you already mastered addition with positive and negative numbers, and this strategy that you convert subtraction to addition means you don't have all that much more to learn. These examples might be computed as follows:
Checkpoint A.1.3.10.
Subsection A.1.4 Multiplying
permalinkMaking sense of multiplication of negative numbers isn't quite so straightforward, but it's possible. Should the product of and be a positive number or a negative number? Remembering that we can view multiplication as repeated addition, we can see this result on a number line:
permalinkFigure A.1.4.11 illustrates that and so it would seem that a positive number times a negative number will always give a negative result. (Note that it would not change things if the negative number came first in the product, since the order of multiplication doesn't affect the result.)
permalinkWhat about the product where both factors are negative? Should the product be positive or negative? If can be seen as adding three times as in Figure A.1.4.12, then it isn't too crazy to interpret as subtracting three times, as in Figure A.1.4.12.
permalinkThis illustrates that and it would seem that a negative number times a negative number always gives a positive result.
permalinkPositive and negative numbers are not the whole story. The number is neither positive nor negative. What happens with multiplication by You can choose to view as adding the number seven times. And you can choose to view as adding the number zero times. Either way, you really added nothing at all, which is the same as adding
Fact A.1.4.13. Multiplication by .
Multiplying any number by results in
Checkpoint A.1.4.14.
Subsection A.1.5 Powers
ΒΆpermalinkFor early sections of this book the only exponents you will see will be the natural numbers : But negative numbers can and will arise as the base of a power.
permalinkAn exponent is a shorthand for how many times to multiply by the base. For example,
permalinkWill the result here be positive or negative? Since we can view as repeated multiplication, and we now understand that multiplying two negatives gives a positive result, this expression can be thought of this way:
permalinkand that lone last negative number will be responsible for making the final product negative.
permalinkMore generally, if the base of a power is negative, then whether or not the result is positive or negative depends on if the exponent is even or odd. It depends on whether or not the factors can all be paired up to βcancelβ negative signs, or if there will be a lone factor left by itself.
permalinkOnce you understand whether the result is positive or negative, for a moment you may forget about signs. Continuing the example, you may calculate that and then since we know is negative, you can report
Warning A.1.5.15. Negative Signs and Exponents.
Expressions like may not mean what you think they mean. What base do you see here? The correct answer is The exponent only applies to the not to So this expression, is actually the same as which is Be careful not to treat as having base That would make it equivalent to which is positive
Checkpoint A.1.5.16.
Subsection A.1.6 Summary
- Addition
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Add two negative numbers: add their positive counterparts and make the result negative.
Add a positive with a negative: find their difference using subtraction, and keep the sign of the dominant number.
- Subtraction
Any subtraction can be converted to addition of the opposite number. For all but the most basic subtractions, this is a useful strategy.
- Multiplication
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Multiply two negative numbers: multiply their positive counterparts and make the result positive.
Multiply a positive with a negative: multiply their positive counterparts and make the result negative.
Multiply any number by the result will be
- Division (not discussed in this section)
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Division by some number is the same as multiplication by its reciprocal. So the multiplication rules can be adopted.
Division of by any nonzero number always results in
Division of any number by is always undefined.
- Powers
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Raise a negative number to an even power: raise the positive counterpart to that power.
Raise a negative number to an odd power: raise the positive counterpart to that power, then make the result negative.
Expressions like mean not
Exercises A.1.7 Exercises
25.
Evaluate the following.
26.
Evaluate the following.
27.
Evaluate the following.
28.
Evaluate the following.
29.
Evaluate the following.
30.
Evaluate the following.
31.
Evaluate the following.
32.
Evaluate the following.
33.
Evaluate the following.
34.
Evaluate the following.
35.
Evaluate the following.
36.
Evaluate the following.
37.
Simplify without using a calculator.
38.
Simplify without using a calculator.
39.
Simplify without using a calculator.
40.
Simplify without using a calculator.
41.
Simplify without using a calculator.
42.
Simplify without using a calculator.
43.
Simplify without using a calculator.
44.
Simplify without using a calculator.
45.
Simplify without using a calculator.
46.
Simplify without using a calculator.
47.
Simplify without using a calculator.
48.
Simplify without using a calculator.
49.
Itβs given that Use this fact to calculate the following without using a calculator.
50.
Itβs given that Use this fact to calculate the following without using a calculator.
51.
Itβs given that Use this fact to calculate the following without using a calculator.
52.
Itβs given that Use this fact to calculate the following without using a calculator.
Applications
53.
Consider the following situation in which you borrow money from your cousin:
On June 1st, you borrowed dollars from your cousin.
On July 1st, you borrowed more dollars from your cousin.
On August 1st, you paid back dollars to your cousin.
On September 1st, you borrowed another dollars from your cousin.
How much money do you owe your cousin now?
54.
Consider the following scenario in which you study your bank account.
On Jan. 1, you had a balance of dollars in your bank account.
On Jan. 2, your bank charged dollar overdraft fee.
On Jan. 3, you deposited dollars.
On Jan. 10, you withdrew dollars.
What is your balance on Jan. 11?
55.
A mountain is feet above sea level. A trench is feet below sea level. What is the difference in elevation between the mountain top and the bottom of the trench?
56.
A mountain is feet above sea level. A trench is feet below sea level. What is the difference in elevation between the mountain top and the bottom of the trench?
Challenge
57.
Select the correct word to make each statement true.
A positive number minus a positive number is
sometimes
always
never
A negative number plus a negative number is
sometimes
always
never
A positive number minus a negative number is
sometimes
always
never
A negative number multiplied by a negative number is
sometimes
always
never