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Section 1.1 Arithmetic

Subsection 1.1.1 Basic Operations

Figure 1.1.1. Basic Operations and Distribution

I assume that you are comfortable with the basic arithmetic operations of addition, subtraction, multiplication and division of numbers. The notations for addition and subtraction are consistent across mathematics.

The notations for multiplication and division, however, tend to change with context. In early context, multiplication is, of course, written with a times sign.

\begin{equation*} 3 \times 7 = 21 \end{equation*}

As our mathematical context grows, however, multiplication is more often written with adjacent symbols. For numbers, which requires parantheses.

\begin{equation*} (3)(7) = 21 \end{equation*}

For variable, writing adjacent variables implies multiplication.

\begin{equation*} ab = a \times b \end{equation*}

Division has the same evolution. In first presentation, division is usually written with a division sign.

\begin{equation*} 12 \div 4 = 3 \end{equation*}

Again as out mathematical context grows, division is often written in fractional notation.

\begin{equation*} \frac{12}{4} = 3 \end{equation*}

These is an important subtlety here: the notation \(\frac{12}{4}\) means two things at the same time: it is a fraction that we can work with and analyse as a fraction (with detail in Chapter 5, but it is also the division \(12 \div 4\text{.}\) When working with this symbol and similar symbols, people switch between these meanings automatically, usually without noticing or announcing the change.

In addition to the four basic arithmetic operations, we also have exponents to indicate iterated multiplication.

\begin{equation*} 2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = (2)(2)(2)(2)(2)(2) = 32 \end{equation*}

Chapter 16 works on the properties and rules of exponents.

Subsection 1.1.2 Distributive Law

Addition and subtraction work well with each other (we can do them in any order). Multiplication and division likewise work well with each other. But we need a rule to describe the interaction between addition/subtraction and multiplication/division. This rule is likely familiar to you in practice, but the terminology might be new (depending on your mathematical background.) We call this the distributive law; more over, we say that multiplication/division distribute over addition/subtraction. This means that the multiplication or division will spread out and apply to each term in the addition or the subtraction.

Here are some examples of the distributve law, using the variety of notations as discussed in the previous section. First, multiplication distributes over addition.

\begin{equation*} (4)(6 + 11) = (4)(6) + (4)(11) = 24 + 44 = 68 \end{equation*}

Second, multiplication distributes over subtraction.

\begin{equation*} 7 \times (12 - 3) = 7 \times 12 - 7 \times 3 = 84 - 21 = 63 \end{equation*}

Third, division distributes over addition.

\begin{equation*} (6 + 4) \div 2 = 6 \div 2 + 4 \div 2 = 3 + 2 + 5 \end{equation*}

Finally, division distributes over subtraction.

\begin{equation*} \frac{1}{7} (21 - 14) = \frac{21}{7} - \frac{14}{7} = 3 - 2 = 1 \end{equation*}

There is a particular case of the distributive law for division when terms are written as fractions. Consider these two examples.

\begin{equation*} \frac{9 + 18}{3} = \frac{9}{3} + \frac{18}{3} = 3 + 6 = 9 \end{equation*}
\begin{equation*} \frac{25 - 10}{5} = \frac{25}{5} - \frac{10}{5} = 5 - 2 = 3 \end{equation*}

This is the distributive law for divisions; the fraction is division and we apply the division to both term in the numerator. Very likely you may have learned this rule by a different name or explanation: something like splitting up the numerators of fractions. In particular, this only applies to numerators. We cannot split up denonminators of fractions. The following statement is NOT true.

\begin{equation*} \frac{1}{2 + 3} \neq \frac{1}{2} + \frac{1}{3} \end{equation*}

The right side is \(\frac{1}{5}\text{.}\) The left side (using common denominator as described in Chapter 5) is \(\frac{5}{6}\text{.}\) These are obviously not the same number.

Subsection 1.1.3 Order of Operations

Figure 1.1.2. Order of Operations

With multiple operations, we have conventions about how to write mathematical expressions so that the operations are clear. Most importantly, we have an order or precedence for operations.

  1. We use parentheses to group pieces of a mathematical expression. Parantheses take precedence over all operations, so they can always be use to be explicit and clear about how we want our expression to be understood.
  2. Sometimes one operations is inside another operation. For example, here is addition inside a square root.
    \begin{equation*} \sqrt{9 + 15} \end{equation*}
    Similarly, here is an exponent and a subtraction inside a fraction (hence inside a division operation).
    \begin{equation*} \frac{50}{2^3+3} \end{equation*}
    Any of these inside pieces have to happen before the outside operation can happen. This is sort of like parenthesis, and we could rewrite these with parentheses to make them more transparent.
    \begin{equation*} \sqrt{(9 + 15)} \end{equation*}
    \begin{equation*} \frac{50}{(2^3+3)} \end{equation*}
    Even without the explicit parentheses, these inside pieces take precedence as if they had parentheses.
  3. Exponents take precedence over the four basic arithmetic operations. Square roots (and other roots) are essentially exponents (as discussed in Chapter 16, so they happen in this step as well.
  4. Multiplication and division take precedence over addition and subtraction. We can treat multiplication and division as one step since we can interchange the order of multiplication and division without problem.
  5. Finally, addition and subtraction have the lowest precedence. Again, we can treat these two in one step because we can interchange the order of additions and subtractions without problem.
\begin{equation*} (5-2)^2 + \frac{21}{7} \end{equation*}
The parentheses take first precedence, so we do the subtraction in the paranethesis.
\begin{equation*} (3)^2 + \frac{21}{7} \end{equation*}
Exponents are next.
\begin{equation*} 9 + \frac{21}{7} \end{equation*}
Division is next.
\begin{equation*} 9 + 3 \end{equation*}
Addition is last.
\begin{equation*} 12 \end{equation*}
\begin{equation*} \frac{(10)(6)(4)}{(2)(5)} \end{equation*}
This is all multiplication and division, so we can do the operations in any order we wish. To demonstrate that the order doesn't matter, I'll do this in a variety of order. First, I'll multiply the numerator.
\begin{equation*} \frac{(60)(4)}{(2)(5)} = \frac{240}{(2)(5) \end{equation*}
Then I'll multiply the denominator.
\begin{equation*} \frac{240}{(2)(5) = \frac{240}{10} \end{equation*}
Then I'll finish with the division.
\begin{equation*} \frac{240}{10} = 24 \end{equation*}
Alternatively, I could have started with 10 divided by 2, which is 5.
\begin{equation*} \frac{(10)(6)(4)}{(2)(5)} = \frac{(5)(6)(4)}{(5)} \end{equation*}
Then I could do 5 divided by 5, which is 1.
\begin{equation*} \frac{(5)(6)(4)}{(5)} = (1)(6)(4) \end{equation*}
Finally, multiplying by 1 doesn't do anything and \(6 \times 4 = 24\text{,}\) so I recover the same answer. There are many more possible orders for all these multiplications and divisions, but all of them will still result in an answer of 24.
\begin{equation*} \frac{\sqrt{5^2 + 11}}{(4)(5) - 14} \end{equation*}
We have two inside pieces: something inside a square root and something inside a fraction (in the denominator). Therefore, we have to do these two pieces before evaluating the square root or the division. First we do the square root. Inside the square root, we have an exponent and addition; we do the exponent first.
\begin{equation*} \frac{\sqrt{25 + 11}}{(4)(5) - 14} \end{equation*}
Then we do the addition.
\begin{equation*} \frac{\sqrt{36}}{(4)(5) - 14} \end{equation*}
The inside of the square root is finished, so we can now do the square root (which has the precedence of an exponent, so it happens before the division).
\begin{equation*} \frac{6}{(4)(5) - 14} \end{equation*}
Now we look to the inside of the division; we can't do the division before the operations in the denominator. The denominator has multiplication and subtraction; we do the multiplication first.
\begin{equation*} \frac{6}{20 - 14} \end{equation*}
Then we do the subtraction.
\begin{equation*} \frac{6}{6} \end{equation*}
Finally, since the inside pieces are finished, we can do the division.
\begin{equation*} \frac{6}{6} = 1 \end{equation*}