In this section we list the properties of real number arithmetic. This is meant to be a succinct, targeted review so we’ll resist the temptation to wax poetic about these axioms and their subtleties and refer the interested reader to a more formal course in Abstract Algebra. There are two primary operations one can perform with real numbers: addition and multiplication. We’ll start with the properties of addition.

• Closure: For all real numbers \(a\) and \(b\), \(a+b\) is also a real number.
• Commutativity: For all real numbers \(a\) and \(b\), \(a+b = b+a\).
• Associativity: For all real numbers \(a\), \(b\) and \(c\), \(a+(b+c) = (a+b)+c\).
• Identity: There is a real number ‘\(0\)’ so that for all real numbers \(a\), \(a+0 = a\).
• Inverse: For all real numbers \(a\), there is a real number \(-a\) such that \(a + (-a) = 0\).
• Definition of Subtraction: For all real numbers \(a\) and \(b\), \(a - b = a + (-b)\).

Next, we give real number multiplication a similar treatment. Recall that we may denote the product of two real numbers \(a\) and \(b\) a variety of ways: \(ab\), \(a \cdot b\), \(a(b)\), \((a)(b)\) and so on. We’ll refrain from using \(a \times b\) for real number multiplication in this text with one notable exception in Definition scientificnotation.

• Closure: For all real numbers \(a\) and \(b\), \(ab\) is also a real number.
• Commutativity: For all real numbers \(a\) and \(b\), \(ab = ba\).
• Associativity: For all real numbers \(a\), \(b\) and \(c\), \(a(bc) = (ab)c\).
• Identity: There is a real number ‘\(1\)’ so that for all real numbers \(a\), \(a \cdot 1 = a\).
• Inverse: For all real numbers \(a \neq 0\), there is a real number \(\frac {1}{a}\) such that \(a \left (\frac {1}{a}\right ) = 1\).
• Definition of Division: For all real numbers \(a\) and \(b \neq 0\), \(a \div b = \frac {a}{b} = a \left (\frac {1}{b}\right )\).

While most students and some faculty tend to skip over these properties or give them a cursory glance at best, ^[1] (Not unlike how Carl approached all the Elven poetry in The Lord of the Rings.) it is important to realize that the properties stated above are what drive the symbolic manipulation in all of Algebra. When listing a tally of more than two numbers, \(1 + 2 + 3\) for example, we don’t need to specify the order in which those numbers are added. Notice though, try as we might, we can add only two numbers at a time and it is the associative property of addition which assures us that we could organize this sum as \((1+2) + 3\) or \(1+(2+3)\). This brings up a note about ‘grouping symbols’. Recall that parentheses and brackets are used in order to specify which operations are to be performed first. In the absence of such grouping symbols, multiplication (and hence division) is given priority over addition (and hence subtraction). For example, \(1 + 2 \cdot 3 = 1+6 = 7\), but \((1+2) \cdot 3 = 3 \cdot 3 = 9\). As you may recall, we can ‘distribute’ the \(3\) across the addition if we really wanted to do the multiplication first: \((1+2) \cdot 3 = 1\cdot 3 + 2 \cdot 3 = 3 + 6 = 9\). More generally, we have the following.

For all real numbers \(a\), \(b\) and \(c\):

• Distributive Property: \(a(b+c) = ab + ac\) and \((a+b)c = ac + bc\).
• Factoring: ^[2] (Or, as Carl calls it, ‘reading the Distributive Property from right to left.’) \(ab+ac = a(b+c)\) and \(ac + bc = (a+b)c\).

It is worth pointing out that we didn’t really need to list the Distributive Property both for \(a(b+c)\) (distributing from the left) and \((a+b)c\) (distributing from the right), since the commutative property of multiplication gives us one from the other. Also, ‘factoring’ is really the same equation as the distributive property, just read from right to left. These are the first of many redundancies in this section, and they exist in this review section for one reason only - in our experience, many students see these things differently so we will list them as such.

It is hard to overstate the importance of the Distributive Property. For example, in the expression \(5(2+x)\), without knowing the value of \(x\), we cannot perform the addition inside the parentheses first; we must rely on the distributive property here to get \(5(2+x) = 5\cdot 2 + 5 \cdot x = 10 + 5x\). The Distributive Property is also responsible for combining ‘like terms’. Why is \(3x + 2x = 5x\)? Because \(3x + 2x = (3+2)x = 5x\).

We continue our review with summaries of other properties of arithmetic, each of which can be derived from the properties listed above. First up are properties of the additive identity \(0\).

Suppose \(a\) and \(b\) are real numbers.

• Zero Product Property: \(ab = 0\) if and only if \(a=0\) or \(b=0\) (or both)

  Note: This not only says that \(0 \cdot a = 0\) for any real number \(a\), it also says that the only way to get an answer of ‘\(0\)’ when multiplying two real numbers is to have one (or both) of the numbers be ‘\(0\)’ in the first place.

• Zeros in Fractions: If \(a \neq 0\), \(\frac {0}{a} = 0 \cdot \left (\frac {1}{a}\right ) = 0\).

  Note: The quantity \(\frac {a}{0}\) is undefined. ^[3] (The expression \(\frac {0}{0}\) is technically an ‘indeterminate form’ as opposed to being strictly ‘undefined’ meaning that with Calculus we can make some sense of it in certain situations. We’ll talk more about this in Chapter RationalFunctions.)

The Zero Product Property drives most of the equation solving algorithms in Algebra because it allows us to take complicated equations and reduce them to simpler ones. For example, you may recall that one way to solve \(x^2+x-6=0\) is by factoring ^[4] (Don’t worry. We’ll review this in due course. And, yes, this is our old friend the Distributive Property!) the left hand side of this equation to get \((x-2)(x+3) = 0\). From here, we apply the Zero Product Property and set each factor equal to zero. This yields \(x-2=0\) or \(x+3=0\) so \(x=2\) or \(x=-3\). This application to solving equations leads, in turn, to some deep and profound structure theorems in Chapter PolynomialFunctions.

Next up is a review of the arithmetic of ‘negatives’. On page 1 we first introduced the dash which we all recognize as the ‘negative’ symbol in terms of the additive inverse. For example, the number \(-3\) (read ‘negative \(3\)’) is defined so that \(3 + (-3) = 0\). We then defined subtraction using the concept of the additive inverse again so that, for example, \(5 - 3 = 5 + (-3)\). In this text we do not distinguish typographically between the dashes in the expressions ‘\(5-3\)’ and ‘\(-3\)’ even though they are mathematically quite different. ^[5] (We’re not just being lazy here. We looked at many of the big publishers’ Precalculus books and none of them use different dashes, either.) In the expression ‘\(5-3\),’ the dash is a binary operation (that is, an operation requiring two numbers) whereas in ‘\(-3\)’, the dash is a unary operation (that is, an operation requiring only one number). You might ask, ‘Who cares?’ Your calculator does - that’s who! In the text we can write \(-3 - 3 = -6\) but that will not work in your calculator. Instead you’d need to type \(^{-}3 - 3\) to get \(-6\) where the first dash comes from the ‘\(+/-\)’ key and the second dash comes from the subtraction key.

Given real numbers \(a\) and \(b\) we have the following.

• Additive Inverse Properties: \(-a = (-1)a\) and \(-(-a) = a\)
• Products of Negatives: \((-a)(-b) = ab\).
• Negatives and Products: \(-ab = -(ab) = (-a)b = a(-b)\).
• Negatives and Fractions: If \(b\) is nonzero, \(-\frac {a}{b} = \frac {-a}{b} = \frac {a}{-b}\) and \(\frac {-a}{-b} = \frac {a}{b}\).
• ‘Distributing’ Negatives: \(-(a+b) = -a-b\) and \(-(a-b) = -a + b = b-a\).
• ‘Factoring’ Negatives: ^[6] (Or, as Carl calls it, reading ‘Distributing’ Negatives from right to left.) \(-a-b = -(a+b)\) and \(b-a = -(a-b)\).

An important point here is that when we ‘distribute’ negatives, we do so across addition or subtraction only. This is because we are really distributing a factor of \(-1\) across each of these terms: \(-(a+b) = (-1)(a+b) = (-1)(a) + (-1)(b) = (-a)+(-b) = -a-b\). Negatives do not ‘distribute’ across multiplication: \(- (2 \cdot 3) \neq (-2)\cdot (-3)\). Instead, \(-(2\cdot 3) = (-2)\cdot (3) = (2) \cdot (-3) = -6\).

The same sort of thing goes for fractions: \(- \frac {3}{5}\) can be written as \(\frac {-3}{5}\) or \(\frac {3}{-5}\), but not \(\frac {-3}{-5}\).

Speaking of fractions, we now review their arithmetic.

Suppose \(a\), \(b\), \(c\) and \(d\) are real numbers. Assume them to be nonzero whenever necessary; for example, when they appear in a denominator.

• Identity Properties: \(a = \frac {a}{1}\) and \(\frac {a}{a} = 1\).
• Fraction Equality: \(\frac {a}{b} = \frac {c}{d}\) if and only if \(ad = bc\).

• Multiplication of Fractions: \(\frac {a}{b} \cdot \frac {c}{d} = \frac {ac}{bd}\). In particular: \(\frac {a}{b} \cdot c = \frac {a}{b} \cdot \frac {c}{1} = \frac {ac}{b}\)

  Note: A common denominator is not required to multiply fractions!

• Division ^[7] (The old ‘invert and multiply’ or ‘fraction gymnastics’ play.) of Fractions: \(\frac {a}{b} \div \frac {c}{d} = \frac {a}{b} \cdot \frac {d}{c} = \frac {ad}{bc}\).

  In particular: \(1 \div \frac {a}{b} = \frac {b}{a}\) and \(\frac {a}{b} \div c = \frac {a}{b} \div \frac {c}{1} = \frac {a}{b} \cdot \frac {1}{c} = \frac {a}{bc}\)

  Note: A common denominator is not required to divide fractions!

• Addition and Subtraction of Fractions: \(\frac {a}{b} \pm \frac {c}{b} = \frac {a \pm c}{b}\).

  Note: A common denominator is required to add or subtract fractions!

• Equivalent Fractions: \(\frac {a}{b} = \frac {ad}{bd}\), since \( \frac {a}{b} = \frac {a}{b} \cdot 1 = \frac {a}{b} \cdot \frac {d}{d} = \frac {ad}{bd}\)

  Note: The only way to change the denominator is to multiply both it and the numerator by the same nonzero value because we are, in essence, multiplying the fraction by \(1\).

• ‘Reducing’ ^[8] (Or ‘Canceling’ Common Factors - this is really just reading the previous property ‘from right to left’.) Fractions: \(\frac {a\cancel {d}}{b\cancel {d}} = \frac {a}{b}\), since \(\frac {ad}{bd} = \frac {a}{b} \cdot \frac {d}{d} = \frac {a}{b} \cdot 1 = \frac {a}{b}\).

  In particular, \(\frac {ab}{b} = a\) since \(\frac {ab}{b} = \frac {ab}{1 \cdot b} = \frac {a \cancel {b}}{1 \cdot \cancel {b}} = \frac {a}{1} = a\) and \(\frac {b-a}{a-b} = \frac {(-1)\cancel {(a-b)}}{\cancel {(a-b)}} = -1\).

  Note: We may only cancel common factors from both numerator and denominator.

Students make so many mistakes with fractions that we feel it is necessary to pause the narrative for a moment and offer you the following examples. Please take the time to read these carefully. In the main body of the text we will skip many of the steps shown here and it is your responsibility to understand the arithmetic behind the computations we use throughout the text. We deliberately limited these examples to “nice” numbers (meaning that the numerators and denominators of the fractions are small integers) and will discuss more complicated matters later. In the upcoming example, we will make use of the Fundamental Theorem of Arithmetic which essentially says that every natural number has a unique prime factorization. Thus ‘lowest terms’ is clearly defined when reducing the fractions you’re about to see.

Of the issues discussed in the previous set of examples none causes students more trouble than simplifying compound fractions. We presented two different methods for simplifying them: one in which we simplified the overall numerator and denominator and then performed the division and one in which we removed the compound nature of the fraction at the very beginning. We encourage the reader to go back and use both methods on each of the compound fractions presented. Keep in mind that when a compound fraction is encountered in the rest of the text it will usually be simplified using only one method and we may not choose your favorite method. Feel free to use the other one in your notes.

Next, we review exponents and their properties. Recall that \(2 \cdot 2 \cdot 2\) can be written as \(2^3\) because exponential notation expresses repeated multiplication. In the expression \(2^3\), \(2\) is called the base and \(3\) is called the exponent. In order to generalize exponents from natural numbers to the integers, and eventually to rational and real numbers, it is helpful to think of the exponent as a count of the number of factors of the base we are multiplying by \(1\). For instance,

From this, it makes sense that

What about \(2^{-3}\)? The ‘\(-\)’ in the exponent indicates that we are ‘taking away’ three factors of two, essentially dividing by three factors of two. So,

We summarize the properties of integer exponents below.

Suppose \(a\) and \(b\) are nonzero real numbers and \(n\) and \(m\) are integers.

• Product Rules: \((ab)^{n} = a^n b^n\) and \(a^n a^m = a^{n+m}\).
• Quotient Rules: \(\left (\frac {a}{b}\right )^n = \frac {a^n}{b^n}\) and \(\frac {a^n}{a^m} = a^{n-m}\).
• Power Rule: \(\left (a^{n}\right )^{m} = a^{nm}\).

• Negatives in Exponents: \(a^{-n} = \frac {1}{a^n}\).

  In particular, \(\left (\frac {a}{b}\right )^{-n} = \left (\frac {b}{a}\right )^{n} = \frac {b^n}{a^n}\) and \(\frac {1}{a^{-n}} = a^{n}\).

• Zero Powers: \(a^{0} = 1\).

  Note: The expression \(0^{0}\) is an indeterminate form. ^[14] (See the comment regarding ‘\(\frac {0}{0}\)’ on page 3.)

• Powers of Zero: For any natural number \(n\), \(0^{n} = 0\).

  Note: The expression \(0^{n}\) for integers \(n \leq 0\) is not defined.

While it is important the state the Properties of Exponents, it is also equally important to take a moment to discuss one of the most common errors in Algebra. It is true that \((ab)^2 = a^2 b^2\) (which some students refer to as ‘distributing’ the exponent to each factor) but you cannot do this sort of thing with addition. That is, in general, \((a+b)^2 \neq a^2 + b^2\). (For example, take \(a= 3\) and \(b = 4\).) The same goes for any other powers.

With exponents now in the mix, we can now state the Order of Operations Agreement.

When evaluating an expression involving real numbers:

1. Evaluate any expressions in parentheses (or other grouping symbols).
2. Evaluate exponents.
3. Evaluate multiplication and division as you read from left to right.
4. Evaluate addition and subtraction as you read from left to right.

We note that there are many useful mnemonic devices for remembering the order of operations. ^[15] (Our favorite is ‘Please entertain my dear auld Sasquatch.’)

For example, \(2 + 3\cdot 4^2 = 2 + 3\cdot 16 = 2 + 48 = 50\). Where students get into trouble is with things like \(-3^2\). If we think of this as \(0 - 3^2\), then it is clear that we evaluate the exponent first: \(-3^2 =0 -3^2 =0 -9 = -9\). In general, we interpret \(-a^n = -\left (a^n\right )\). If we want the ‘negative’ to also be raised to a power, we must write \((-a)^n\) instead. To summarize, \(-3^2 = -9\) but \((-3)^2 = 9\).

Of course, many of the ‘properties’ we’ve stated in this section can be viewed as ways to circumvent the order of operations. We’ve already seen how the distributive property allows us to simplify \(5(2+x)\) by performing the indicated multiplication before the addition that’s in parentheses. Similarly, consider trying to evaluate \(2^{30172}\cdot 2^{-30169}\). The Order of Operations Agreement demands that the exponents be dealt with first, however, trying to compute \(2^{30172}\) is a challenge, even for a calculator. One of the Product Rules of Exponents, however, allow us to rewrite this product, essentially performing the multiplication first, to get: \(2^{30172-30169} = 2^{3} = 8\).

Let’s take a break and enjoy another example.

One of the places where the properties of exponents play an important role is in the use of Scientific Notation. The basis for scientific notation is that since we use decimals (base ten numerals) to represent real numbers, we can adjust where the decimal point lies by multiplying by an appropriate power of 10. This allows scientists and engineers to focus in on the ‘significant’ digits ^[16] (Awesome pun!) of a number - the nonzero values - and adjust for the decimal places later. For instance, \(-621 = -6.21 \times 10^2\) and \(0.023 = 2.3 \times 10^{-2}\). Notice here that we revert to using the familiar ‘\(\times \)’ to indicate multiplication. ^[17] (This is the ‘notable exception’ we alluded to earlier.) In general, we arrange the real number so exactly one non-zero digit appears to the left of the decimal point. We make this idea precise in the following:

On calculators, scientific notation may appear using an ‘E’ or ‘EE’ as opposed to the \(\times \) symbol. For instance, while we will write \(6.02 \times 10^{23}\) in the text, the calculator may display \(6.02\, \text {E} \, 23\) or \(6.02\, \text {EE} \, 23\).

We close our review of real number arithmetic with a discussion of roots and radical notation. Just as subtraction and division were defined in terms of the inverse of addition and multiplication, respectively, we define roots by undoing natural number exponents.

The reasons for the added stipulations for even-indexed roots in Definition principalnthrootdefn can be found in the Properties of Negatives. First, for all real numbers, \(x^{\text {even power}} \geq 0\), which means it is never negative. Thus if \(a\) is a negative real number, there are no real numbers \(x\) with \(x^{\text {even power}} = a\). This is why if \(n\) is even, \(\sqrt [n]{a}\) only exists if \(a \geq 0\). The second restriction for even-indexed roots is that \(\sqrt [n]{a} \geq 0\). This comes from the fact that \(x^{\text {even power}} = (-x)^{\text {even power}}\), and we require \(\sqrt [n]{a}\) to have just one value. So even though \(2^{4} = 16\) and \((-2)^{4} = 16\), we require \(\sqrt [4]{16} = 2\) and ignore \(-2\).

Dealing with odd powers is much easier. For example, \(x^3 = -8\) has one and only one real solution, namely \(x = -2\), which means not only does \(\sqrt [3]{-8}\) exist, there is only one choice, namely \(\sqrt [3]{-8} = -2\). Of course, when it comes to solving \(x^{5213} = -117\), it’s not so clear that there is one and only one real solution, let alone that the solution is \(\sqrt [5213]{-117}\). Such pills are easier to swallow once we’ve thought a bit about such equations graphically, ^[18] (See Chapter PolynomialFunctions.) and ultimately, these things come from the completeness property of the real numbers mentioned earlier.

We list properties of radicals below as a ‘theorem’ as opposed to a definition since they can be justified using the properties of exponents.

The proof of Theorem radicalprops is based on the definition of the principal \(\text {n}^{\text {th}}\) root and the Properties of Exponents. To establish the product rule, consider the following. If \(n\) is odd, then by definition \(\sqrt [n]{ab}\) is the unique real number such that \((\sqrt [n]{ab})^{n} = ab\). Given that \(( \sqrt [n]{a} \, \sqrt [n]{b})^n = (\sqrt [n]{a})^n (\sqrt [n]{b})^n = ab\) as well, it must be the case that \(\sqrt [n]{ab} = \sqrt [n]{a} \, \sqrt [n]{b}\). If \(n\) is even, then \(\sqrt [n]{ab}\) is the unique non-negative real number such that \((\sqrt [n]{ab})^{n} = ab\). Note that since \(n\) is even, \(\sqrt [n]{a}\) and \(\sqrt [n]{b}\) are also non-negative thus \(\sqrt [n]{a}\sqrt [n]{b} \geq 0\) as well. Proceeding as above, we find that \(\sqrt [n]{ab} = \sqrt [n]{a} \, \sqrt [n]{b}\). The quotient rule is proved similarly and is left as an exercise. The power rule results from repeated application of the product rule, so long as \(\sqrt [n]{a}\) is a real number to start with. ^[19] (Otherwise we’d run into an interesting paradox. See Section AppCmpNums.) We leave that as an exercise as well.

We pause here to point out one of the most common errors students make when working with radicals. Obviously \(\sqrt {9} = 3\), \(\sqrt {16} = 4\) and \(\sqrt {9 + 16} = \sqrt {25} = 5\). Thus we can clearly see that \(5 = \sqrt {25} = \sqrt {9 + 16} \neq \sqrt {9} + \sqrt {16} = 3 + 4 = 7\) because we all know that \(5 \neq 7\). The authors urge you to never consider ‘distributing’ roots or exponents. It’s wrong and no good will come of it because in general \(\sqrt [n]{a+b} \neq \sqrt [n]{a} + \sqrt [n]{b}\).

Since radicals have properties inherited from exponents, they are often written as such. We define rational exponents in terms of radicals in the box below.

It would make life really nice if the rational exponents defined in Definition rationalexponentdefn had all of the same properties that integer exponents have as listed on page 10 - but they don’t. Why not? Let’s look at an example to see what goes wrong. Consider the Product Rule which says that \((ab)^{n} = a^{n}b^{n}\) and let \(a = -16\), \(b = -81\) and \(n = \frac {1}{4}\). Plugging the values into the Product Rule yields the equation \(((-16)(-81))^{1/4} = (-16)^{1/4}(-81)^{1/4}\). The left side of this equation is \(1296^{1/4}\) which equals \(6\) but the right side is undefined because neither root is a real number. Would it help if, when it comes to even roots (as signified by even denominators in the fractional exponents), we ensure that everything they apply to is non-negative? That works for some of the rules - we leave it as an exercise to see which ones - but does not work for the Power Rule.

Consider the expression \(\left (a^{2/3}\right )^{3/2}\). Applying the usual laws of exponents, we’d be tempted to simplify this as \(\left (a^{2/3}\right )^{3/2} = a^{\frac {2}{3} \cdot \frac {3}{2}} = a^{1} = a\). However, if we substitute \(a=-1\) and apply Definition rationalexponentdefn, we find \((-1)^{2/3} = \left (\sqrt [3]{-1}\right )^2 = (-1)^2 = 1\) so that \(\left ((-1)^{2/3}\right )^{3/2} = 1^{3/2} = \left (\sqrt {1}\right )^3 = 1^3 = 1\). Thus in this case we have \(\left (a^{2/3}\right )^{3/2} \neq a\) even though all of the roots were defined. It is true, however, that \(\left (a^{3/2}\right )^{2/3} = a\) and we leave this for the reader to show. The moral of the story is that when simplifying powers of rational exponents where the base is negative or worse, unknown, it’s usually best to rewrite them as radicals. ^[21] (Much to Jeff’s chagrin. He’s fairly traditional and therefore doesn’t care much for radicals.)

We close this section with a note about simplifying. In the preceding examples we used “nice” numbers because we wanted to show as many properties as we could per example. This then begs the question “What happens when the numbers are not nice?” Unfortunately, the answer is “Not much simplifying can be done.” Take, for example,

Sadly, that’s as good as it gets.