Survey of Algebra and Probability: MAT 101

Subsection 1.A.1 Fractions

Here we will briefly review operations on fractions. A fraction can be used to represent parts of a whole. The fraction bar shown below represents the fraction \(\frac{1}{3}\) since one of three equal parts of the bar is shaded.

If we think of the bar as the number \(1\text{,}\) the fraction \(\frac{1}{3}\) also represents the operation of dividing one into three parts, \(1 \div 3\text{.}\) We may use the representations \(\frac{a}{b}\) and \(a \div b\) interchangeably. The upper number of a fraction is called the numerator and the bottom value is called the denominator of the fraction.

We could have divided the bar into six equal parts with two shaded. In this case, the original segments are divided into two equal pieces.

Comparing the two bars, the area in the shaded parts are equal. The only difference is that the second bar is divided into twice as many pieces. There are many ways to express the same quantity, for example: \(\frac{1}{3} = \frac{2}{6} = \frac{4}{12}\text{.}\) If we drew the bar for \(\frac{4}{12}\text{,}\) the entire bar and the shaded area would be divided into four times as many pieces as the original bar. If we further subdivide the original segments, the shaded area is subdivided in the same way. Different expressions for the same fraction may be obtained by multipying numerator and denominator by the same number.

When working with fractions, we like our final answers to be reduced. This means numbers in the numerator and denominator are as small as possible. Just as multiplying top and bottom of a fraction by the same number produces an equivalent fraction, dividing top and bottom by the same number does not change the value. Visualize this as reducing the number of subdivisions. One way to accomplish reduced form is to divide the numerator and denominator by common divisors until no common divisor remains.

The previous Example 1.A.2could have been done in one step by dividing both numerator and denominator by \(12.\) We also could have divided by \(2\) twice and then divided by \(3\) once (in any order). It is not important which method we use as long as we continue reducing our fraction until it cannot be reduced any further. Another way to reduce the fraction is Example 1.A.2is given in the next example.

Consider the fraction \(\frac{12}{12}\) in the example above. If we write it in division form, it is clear that: \(12 \div 12 = 1\text{.}\) For any nonzero number \(a\text{,}\)

Zero is excluded because if we consider the expression \(\frac{0}{0}\) or \(0 \div 0\) we see there are many numbers that multiply zero and give zero: \(1 \cdot 0 = 0\text{,}\) \(2 \cdot 0 = 0\text{,}\) \(3 \cdot 0 = 0\) and so on. We say the expression is not defined since there is no way to determine one answer. Similarily, for any nonzero number \(n\text{,}\) \(\frac{n}{0}\) or \(n \div 0\text{,}\) is not defined because there is no number that can multiply zero to get a nonzero number. On the other hand, a fraction with a zero in the numerator is always equal to zero. For example,

Note that any integer can be represented as a fraction. For example, \(7\) can be written as the fraction \(\frac{7}{1}\) which represents seven one's. However, \(7\) is considered “reduced form”.

Subsection 1.A.2 Multiplication and Division of Fractions

We multiply fractions by multiplying their numerators together and their denominators together. This process is a shortcut for what it means to multiply fractions. We give a demonstration in three steps to illustrate.

First, we consider multiplying a fraction by a whole number:

Here, we add \(\frac{1}{4}\) to itself five times.

Second, we consider multiplying the simple fractions: \(\frac{1}{4} \cdot \frac{1}{2}\text{.}\) Instead of summing a number of one-fourths we want half of one-fourth.

Finally, we put the two ideas together. Using the first idea, we know \(\frac{1}{2} \cdot 5\) is adding five one-halfs or \(\frac{5}{2}\text{.}\) We can rewrite the multiplication as follows:

Now, we do not need to do all of these steps to multiply fractions. We only showed them to demonstrate why the process of multipying fractions works. What you need to remember is:

Note that to be consistent, we could rewrite integers as fractions, \(\frac{1}{4} \cdot 5 = \frac{1}{4} \cdot \frac{5}{1}= \frac{5}{4},\) but it is not necessary to rewrite the \(5\) to do the multiplication.

We can reduce fractions before we multiply. We can either reduce "vertically" (within a single fraction) or "diagonally" (across several fractions), as long as we divide the same number from the numerator and from the denominator.

Dividing fractions is very similar to multiplying with one extra step. We first take the reciprocal of the second fraction, and then multiply. Note that the reciprocal of a fraction, \(\frac{a}{b}\text{,}\) is made by switching the numerator and denominator of the fraction to get \(\frac{b}{a}\text{.}\) As with multiplication, this process is a short-cut for a longer sequence of steps. We demonstrate the full sequence of steps below. Note that we use the fraction notation for division with \(\frac{3}{7} \div \frac{2}{5} = \frac{\frac{3}{7}}{\frac{2}{5}}\text{.}\)

We know that if we multiply the numerator and denominator by the same (non-zero) number, the value of the fraction is unchanged. We multiply both by \(\frac{5}{2}\text{.}\)

Multiply the bottom fractions to get

Since any number divided by one is itself, our expression becomes

This shows we may rewrite division by fractions into multiplication. The result of the division is

It is not necessary to go thru the entire process shown above to divide fractions. We may go straight to multiplying by the reciprocal of the denominator as summarized below.

Subsection 1.A.3 Addition and Subtraction of Fractions

If both fractions have the same denominator, then the fractions have with the same number of subdivisions and we sum or subtract the amount in the numerator. We call fractions with the same denominator like fractions.

Although \(\dfrac{5}{4}\) can be written as the mixed number \(1\dfrac{1}{4}\text{,}\) we will always use the improper fraction form (the numerator is larger than the denominator).

If the denominators are not equal, then the sum is of subintervals of different lengths. For example, if we add \(\frac{1}{3}\) to \(\frac{1}{2},\) as shown below, the result is not in terms of halves or thirds.

To add or subtract fractions they need to have the same denominator, in other words, a common denominator.

Subsection 1.A.4 Common Denominators

The common denominator will be evenly divisible by both denominators. One way to determine the smallest such number is to multiply the larger denominator by integers until the result is also divisible by the smaller denominator. The smallest possible common divisor is called the least common denominator or LCD. Once we determine the LCD, we adjust each fraction so that it's denominator is the LCD.

To summarize:

Subsection 1.A.5 Order of Operations With Fractions

When simplifying mathematical expressions, a fraction bar may act as a set of grouping symbols like parentheses. The entire numerator and the entire denominator of a fraction must be evaluated before we reduce the fraction. Think of the fraction bar like this: