There are special things to consider when adding, subtracting, multiplying, dividing, and raising to powers, when negative numbers are involved. This section reviews those arithmetic operations with negative numbers.
Is it valid to subtract a large number from a smaller one? It may be hard to imagine what it would mean physically to subtract \(3\) cars from your garage if you only have \(1\) car there in there in the first place. But mathematics gives meaning to expressions like \(1-3\) using signed numbers.
You’ve probably seen signed numbers used to describe the temperature of very cold things. Most people on Earth use the Celsius scale for temperature. If you’re not familiar with the Celsius temperature scale, think about these examples:
Figure 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 \(0\) are called positive numbers and values to the left of \(0\) are called negative numbers.
WarningA.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. We must be able to identify when a “minus” sign means to subtract and when it means to negate. Here are some examples.
\(-13\) has one negative sign and no subtraction sign.
To adding two numbers with the same sign you can (at first) ignore their signs, and add the two numbers as if they were positive. Then make sure your result is either positive or negative, depending on what the sign was.
If you needed to add \(-18\) and \(-7\text{,}\) note that both are negative. Maybe you have this expression in front of you:
\begin{equation*}
-18+-7
\end{equation*}
That “plus minus” is awkward, and in this book you are more likely to see this expression:
\begin{equation*}
-18+(-7)
\end{equation*}
with extra parentheses. Since both terms are negative, we can add \(18\) and \(7\) to get \(25\) but realize that our final result should be negative. So our result is \(-25\text{:}\)
This approach works because adding numbers is like having two people tugging on a rope, with strength indicated by each number. In Example 5 we have two people pulling to the left, one with strength \(18\) and the other with strength \(7\text{.}\) Their forces combine to pull left with strength \(25\text{,}\) giving us our total of \(-25\text{,}\) as illustrated in Figure 6.
If we are adding two numbers that have opposite signs, then the two people are tugging the rope in opposing directions. If either of them is using more strength than the other, then overall there will be a net pull in the stronger person’s direction. And the overall pull on the rope will be the difference of the two strengths. This is illustrated in Figure 7.
We have one number of each sign, with sizes \(15\) and \(12\text{.}\) Their difference is \(3\text{.}\) But of the two numbers, the negative number is stronger. So the result from adding these is also negative: \(-3\text{.}\)
We have one number of each sign, with sizes \(200\) and \(100\text{.}\) Their difference is \(100\text{.}\) But of the two numbers, the positive number is stronger. So the result from adding these is also positive: \(100\text{.}\)
We have one number of each sign, with sizes \(12.8\) and \(20\text{.}\) Their difference is \(7.2\text{.}\) But of the two numbers, the negative number is stronger. So the result from adding these is also negative: \(-7.2\text{.}\)
The two numbers have opposite sign, so we subtract \(9-1=8\text{.}\) Of the two numbers being added, the positive is larger, so the result should positive as well: \(8\text{.}\)
The two numbers have opposite sign, so we subtract \(123-100=23\text{.}\) Of the two numbers being added, the negative is larger, so the result should be negative: \(-23\text{.}\)
The two numbers have opposite sign, so we can subtract \(81.53-34.67=46.86\text{.}\) Of the two numbers being added, the positive is larger, so the result should be positive: \(46.86\text{.}\)
Subtracting a small positive number from a larger number, such as \(18-5\text{,}\) is a skill you are familiar with. Subtraction can also be done where a small positive number subtracts a larger number, or where one or both numbers are negative. Subtracting with negative numbers can cause confusion, and to avoid that confusion, it may help to think of subtraction as adding the opposite number.
This strategy will reduce subtraction to addition. So if you are already comfortable adding positive and negative numbers, subtraction becomes just as familiar. These examples show how it is done:
We can change this to \(32+(-50)\text{.}\) Two numbers are added, and the larger one is negative. So we find the difference \(50-32=18\text{,}\) but the final result must be negative: \(-18\text{.}\)
We can rewrite this as \(-5.9+3.1\text{.}\) Now it is the sum of two numbers of opposite sign, so we find the difference \(5.9-3.1=2.8\text{.}\) We were adding two numbers where the larger one was negative. So the final result should also be negative: \(-2.8\text{.}\)
Since we are subtracting a positive number from a negative number, the result should be an even more negative number. We can add \(12.04+17.2\) to get \(29.24\text{,}\) but our final answer should be the opposite, \(-29.24\text{.}\)
Multiplication with negative numbers is possible too. We can view multiplication as repeated addition. For example \(3\cdot7=7+7+7\text{.}\) We can do the same when there is a negative number in the product. Figure 11 represents \(3\cdot(-7)\text{.}\)
What about the product \(-3\cdot(-7)\text{,}\) where both factors are negative? Should the result be positive or negative? If \(3\cdot(-7)\) can be seen as adding\(-7\) three times as in Figure 11, then it isn’t too crazy to interpret \(-3\cdot(-7)\) as subtracting\(-7\) three times. Or in other words, as adding\(7\) three times. This is illustrated in Figure 12.
Positive and negative numbers are not the whole story. The number \(0\) is neither positive nor negative. What happens with multiplication by \(0\text{?}\) You can choose to view \(7\cdot0\) as adding the number \(0\) seven times. And you can choose to view \(0\cdot7\) as adding the number \(7\) zero times. Either way, the result is \(0\text{.}\)
Negative numbers can arise as the base of a power. An exponent is shorthand for how many times to multiply the base together. For example, \((-2)^5\) means
Will the result here be positive or negative? Since we can view \((-2)^5\) as repeated multiplication, and since multiplying two negatives gives a positive result, this expression can be thought of this way:
More 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 “cancel” negative signs, or if there will be a lone negative factor left unpaired.
Once you understand whether the result is positive or negative, for a moment you may forget about signs. Returning to the example, you could calculate that \(2^5=32\text{,}\) and then since we separately know that \((-2)^5\) should be negative, you can conclude:
Expressions like \(-3^4\) may not mean what you think they mean. What base do you see here? The correct answer is \(3\text{.}\) The exponent \(4\) only applies to the \(3\text{,}\) not to \(-3\text{.}\) So this expression, \(-3^4\text{,}\) is actually the same as \(-\mathopen{}\left(3^4\right)\mathclose{}\text{,}\) which is \(-81\text{.}\) Be careful not to treat \(-3^4\) as having base \(-3\text{.}\) That would make it equivalent to \((-3)^4\text{,}\) which is positive\(81\text{.}\)
A mountain is \(1100\) feet above sea level. A trench is \(360\) feet below sea level. What is the difference in elevation between the mountain top and the bottom of the trench?
A mountain is \(1200\) feet above sea level. A trench is \(420\) feet below sea level. What is the difference in elevation between the mountain top and the bottom of the trench?
