ORCCA Arithmetic with Negative Numbers

Section 1.1 Arithmetic with Negative Numbers

Adding, 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 1.1.1 Signed 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 \(8\) cars from your garage if you only have \(1\) car in there in the first place. Nevertheless, mathematics has found a way to give meaning to expressions like \(1-8\) using signed numbers.

In 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:

a number line, with values in degrees Celsius; marks average temperature at North Pole in winter at -40; marks cold winter night at -10; marks water freezes at 0; marks cool autumn day at 13; marks human body temperature at 37 — Figure 1.1.1. Number line with interesting Celsius temperatures

Figure 1.1.1 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.

Warning 1.1.2. 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 “minus” sign means “subtract” or means “negative.” 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.

\(-13\) has one negative sign and no subtraction sign.
\(20-13\) has no negative signs and one subtraction sign.
\(-20-13\) has a negative sign and then a subtraction sign.
\((-20)(-13)\) has two negative signs and no subtraction sign.

Checkpoint 1.1.3.

Identify “minus” signs.

Subsection 1.1.2 Adding

An 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 1.1.4. Add Two Negative Numbers.

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\text{,} \end{equation*}

but that “plus minus” is awkward, and in this book you are more likely to have this expression:

\begin{equation*} -18+(-7) \end{equation*}

with extra parentheses. (How many subtraction signs do you see? How many negative signs?)

Since both our terms are negative, we can add \(18\) and \(7\) to get \(25\) and immediately realize that our final result should be negative. So our result is \(-25\text{:}\)

\begin{equation*} -18+(-7)=-25 \end{equation*}

This 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 1.1.4 we have two people pulling to the left, one with strength \(18\text{,}\) 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 1.1.5.

If 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 1.1.6.

Figure 1.1.5. Working together

Figure 1.1.6. Working in opposition

Example 1.1.7. Adding One Number of Each Sign.

Here are four examples of addition where one number is positive and the other is negative.

\(-15+12\)

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 dominates. So the result from adding these is \(-3\text{:}\)

\begin{equation*} -15+12=-3 \end{equation*}
\(200+(-100)\)

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 dominates. So the result from adding these is \(100\text{:}\)

\begin{equation*} 200+(-100)=100 \end{equation*}
\(12.8+(-20)\)

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 dominates. So the result from adding these is \(-7.2\text{:}\)

\begin{equation*} 12.8+(-20)=-7.2 \end{equation*}
\(-87.3+87.3\)

We have one number of each sign, both with size \(87.3\text{.}\) The opposing forces cancel each other, leaving a result of \(0\text{.}\)

Checkpoint 1.1.8.

Subsection 1.1.3 Subtracting

Perhaps you can handle a subtraction such as \(18-5\text{,}\) 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:	\(12-30\)	\(12+(-30)\)
Subtracting from a negative number:	\(-8.1-17\)	\(-8.1+(-17)\)
Subtracting a negative number:	\(42-(-23)\)	\(42+23\)

The 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:

\begin{align*} 12-30\amp=12+(-30)\\ \amp=-18 \end{align*}

\begin{align*} -8.1-17\amp=-8.1+(-17)\\ \amp=-25.1 \end{align*}

\begin{align*} 42-(-23)\amp=42+23\\ \amp=65 \end{align*}

Checkpoint 1.1.9.

Subsection 1.1.4 Multiplying

Making sense of multiplication of negative numbers isn't quite so straightforward, but it's possible. Should the product of \(3\) and \(-7\) 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:

Figure 1.1.10. Viewing \(3\cdot(-7)\) as repeated addition

Figure 1.1.10 illustrates that \(3\cdot(-7)=-21\) 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.)

What about the product \(-3\cdot(-7)\text{,}\) where both factors are negative? Should the product be positive or negative? If \(3\cdot(-7)\) can be seen as adding \(-7\) three times as in Figure 1.1.11, then it isn't too crazy to interpret \(-3\cdot(-7)\) as subtracting \(-7\) three times, as in Figure 1.1.11.

Figure 1.1.11. Viewing \(-3\cdot(-7)\) as repeated subtraction

This illustrates that \(-3\cdot(-7)=21\) and it would seem that a negative number times a negative number always gives a positive result.

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, you really added nothing at all, which is the same as adding \(0\text{.}\)

Fact 1.1.12. Multiplication by \(0\).

Multiplying any number by \(0\) results in \(0\text{.}\)

Checkpoint 1.1.13.

Subsection 1.1.5 Division

Division by some number is the same as multiplication by its reciprocal. For example, dividing a number by \(2\) is the same as multiplying by the reciprocal of \(2\text{,}\) which is \(\frac{1}{2}\text{.}\) Thus, when dividing numbers, including negative numbers, the same rules we use for multiplication can be applied when determining the resulting sign.

Example 1.1.14. Examples of Division with Negative Numbers.

Here are four examples of division where at least one number is negative.

\(-36\div12\)

We have one number of each sign. Recall, that we learned in the previous section, the rule for multiplication is that a negative times a positive always results in a negative. Since the same rules for multiplication can be applied to division, when determining the resulting sign, to calculate this answer, we simply divide \(36\) by \(12\text{,}\) which gives us \(3\) and then since the numbers have opposite sign, we know the final result is negative. Thus,

\begin{equation*} -36\div12=-3 \end{equation*}
\(200\div(-100)\)

Again, we have one number of each sign, so the result will be negative.

\begin{equation*} 200\div(-100)=-2 \end{equation*}
\(-\frac{15}{2}\div-25\)

This time the numbers are both negative. Recall, from multiplication, the rule is that if the numbers are both negative, the result will be positive. Since we are working with fractions in this case, it is clear why the same rules for multiplication are applied to division: because division is the same as multiplying by the reciprocal of a number.

\begin{align*} -\frac{15}{2}\div-25\amp=-\frac{15}{2}\cdot-\frac{1}{25}\\ \amp=\frac{3}{10} \end{align*}
\(\frac{-96}{-52}\)

Both numbers are negative, in this case, so the final result will be positive. However, \(96\) cannot be evenly divided by \(52\text{,}\) so we simply reduce the fraction and then simplify the signs by making the final result positive.

\begin{equation*} \frac{-96}{-52}=\frac{24}{13} \end{equation*}

Checkpoint 1.1.15.

Subsection 1.1.6 Powers

For early sections of this book the only exponents you will see will be the natural numbers: \(\{1, 2, 3, \ldots\}\text{.}\) But negative numbers can and will arise as the base of a power.

An exponent is a shorthand for how many times to multiply by the base. For example,

\begin{equation*} (-2)^5\text{ means }\overbrace{(-2)\cdot(-2)\cdot(-2)\cdot(-2)\cdot(-2)}^{5\text{ instances}} \end{equation*}

Will the result here be positive or negative? Since we can view \((-2)^5\) as repeated multiplication and we now understand that multiplying two negatives gives a positive result, this expression can be thought of this way:

\begin{equation*} \underbrace{\underbrace{(-2)\cdot(-2)}_{\text{positive}}\cdot\underbrace{(-2)\cdot(-2)}_{\text{positive}}}_{\text{positive}}\cdot(-2) \end{equation*}

and that lone last negative number will be responsible for making the final product negative.

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 factor left by itself.

Once you understand whether the result is positive or negative, for a moment you may forget about signs. Continuing the example, you may calculate that \(2^5=32\) and then, since we know \((-2)^5\) is negative, you can report

\begin{equation*} (-2)^5=-32 \end{equation*}

Warning 1.1.16. Negative Signs and Exponents.

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 \(81\text{.}\)

Checkpoint 1.1.17.

Subsection 1.1.7 Summary

Addition

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

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 \(0\text{:}\) the result will be \(0\text{.}\)

Division

Division by some number is the same as multiplication by its reciprocal. So the multiplication rules can be adopted.

Division of \(0\) by any nonzero number always results in \(0\text{.}\)

Division of any number by \(0\) is always undefined.

Powers

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 \(-2^4\) mean \(-\mathopen{}\left(2^4\right)\mathclose{}\text{,}\) not \((-2)^4\text{.}\)

Exercises 1.1.8 Exercises

1.

Add the following.

\(-8+(-3)\)
\(-4+(-5)\)
\(-1+(-9)\)

2.

Add the following.

\(-8+(-1)\)
\(-6+(-7)\)
\(-1+(-7)\)

3.

Add the following.

\(1+(-8)\)
\(9+(-1)\)
\(7+(-7)\)

4.

Add the following.

\(2+(-10)\)
\(5+(-2)\)
\(7+(-7)\)

5.

Add the following.

\(-6+2\)
\(-3+8\)
\(-4+4\)

6.

Add the following.

\(-8+3\)
\(-4+10\)
\(-4+4\)

7.

Add the following.

\(-51+(-90)\)
\(-69+83\)
\(65+(-71)\)

8.

Add the following.

\(-41+(-31)\)
\(-31+58\)
\(64+(-27)\)

9.

Subtract the following.

\(4-8\)
\(6-2\)
\(4-14\)

10.

Subtract the following.

\(5-6\)
\(8-4\)
\(4-19\)

11.

Subtract the following.

\(-1-4\)
\(-10-4\)
\(-6-6\)

12.

Subtract the following.

\(-5-2\)
\(-8-3\)
\(-6-6\)

13.

Subtract the following.

\(-2-(-6)\)
\(-10-(-1)\)
\(-4-(-4)\)

14.

Subtract the following.

\(-2-(-8)\)
\(-6-(-2)\)
\(-4-(-4)\)

15.

Perform the given addition and subtraction.

\({-16-4+\left(-4\right)}\)
\({6-\left(-15\right)+\left(-17\right)}\)

16.

Perform the given addition and subtraction.

\({-15-10+\left(-9\right)}\)
\({4-\left(-15\right)+\left(-12\right)}\)

17.

Perform the given addition and subtraction.

\({-13-7+\left(-5\right)}\)
\({1-\left(-15\right)+\left(-17\right)}\)

18.

Perform the given addition and subtraction.

\({-12-3+\left(-1\right)}\)
\({8-\left(-15\right)+\left(-12\right)}\)

19.

Multiply the following.

\((-8)\cdot(-1)\)
\((-6)\cdot4\)
\(7\cdot(-5)\)
\((-6)\cdot0\)

20.

Multiply the following.

\((-8)\cdot(-2)\)
\((-4)\cdot3\)
\(7\cdot(-2)\)
\((-5)\cdot0\)

21.

Multiply the following.

\((-3)\cdot(-6)\cdot(-5)\)
\(7\cdot(-8)\cdot(-4)\)
\((-86)\cdot(-64)\cdot0\)

22.

Multiply the following.

\((-3)\cdot(-4)\cdot(-3)\)
\(5\cdot(-8)\cdot(-2)\)
\((-84)\cdot(-52)\cdot0\)

23.

Multiply the following.

\(\displaystyle{ (-1)(-2)(-1)(-2) }\)
\(\displaystyle{ (-1)(-3)(-3)(-3) }\)

24.

Multiply the following.

\(\displaystyle{ (-2)(-1)(-2)(-1) }\)
\(\displaystyle{ (-3)(-3)(1)(-1) }\)

25.

Evaluate the following.

\(\frac{-54}{-6}\)
\(\frac{36}{-4}\)
\(\frac{-24}{6}\)

26.

Evaluate the following.

\(\frac{-30}{-5}\)
\(\frac{36}{-6}\)
\(\frac{-48}{6}\)

27.

Evaluate the following.

\(\frac{-4}{-1}\)
\(\frac{5}{-1}\)
\(\frac{170}{-170}\)
\(\frac{-18}{-18}\)
\(\frac{8}{0}\)
\(\frac{0}{-8}\)

28.

Evaluate the following.

\(\frac{-3}{-1}\)
\(\frac{9}{-1}\)
\(\frac{110}{-110}\)
\(\frac{-10}{-10}\)
\(\frac{8}{0}\)
\(\frac{0}{-4}\)

29.

Evaluate the following.

\((-9)^{2}\)
\(-4^{2}\)

30.

Evaluate the following.

\((-7)^{2}\)
\(-6^{2}\)

31.

Evaluate the following.

\((-3)^{3}\)
\(-1^{3}\)

32.

Evaluate the following.

\((-2)^{3}\)
\(-4^{3}\)

33.

Evaluate the following.

\(4^{2}\)
\(2^{3}\)
\((-4)^{2}\)
\((-3)^{3}\)

34.

Evaluate the following.

\(5^{2}\)
\(4^{3}\)
\((-3)^{2}\)
\((-5)^{3}\)

35.

Evaluate the following.

\(1^{10}\)
\((-1)^{11}\)
\((-1)^{12}\)
\(0^{20}\)