Algebra Intro 9: Fractions, Reciprocals, and Properties of Division

Dividing six by two

Now that fractions and rational numbers have been introduced, let’s explore what they do for us a bit

Fractions And Rational Numbers

A rational number is one that can be represented by the ratio of two integers.

A fraction is a number that is written as the ratio of two quantities.

This means that the number 5 is a rational number because it can be represented as


which is a ratio of two integers – and one of many such possible ratios.

5 is not a fraction because, while it can be written as the ratio of two quantities, it is not written that way in this case.

\dfrac{10}{2} is a fraction because it is written as a ratio of two quantities.

Based on the above definitions, all rational numbers can be represented using fractions, however not all fractions will represent rational numbers.  For example,


is a fraction, but it is not a rational number because it cannot be represented by the ratio of two integers.

Interpreting Fractions

A fraction is a ratio of two quantities.  But what does it mean?  There are several ways to interpret fractions.

The simplest is as a division problem:


means “divide six into two equal portions: how big will each portion be?” Or “what quantity must I multiply by two to obtain six”?  The answer will be three:

Or you may interpret a fraction as a grouping situation.  The above fraction represents 6 items arranged in groups of 2.  How many such groups are there?  Three:

A similar approach is to interpret a fraction as parts of a whole.  The fraction above, as well as the diagram immediately above, represent six parts (the numerator) of a type where two parts make up a “whole” (the denominator).  If two parts make up a “whole”, then I can create three “wholes” from the six parts I have, and the fraction therefore represents the quantity three.

“Proper” And “Improper” Fractions

A “proper” fraction has a numerator that is closer to zero than its denominator.  Since this means that there are fewer parts than needed to make “a whole”, the decimal value of such a fraction will be between negative one and one.

An “improper” fraction has a numerator that is further from zero than its denominator, so it represents a number that is further away from zero than negative one or positive one.

Why do we bother distinguishing between proper and improper fractions? One reason is that an “improper” fraction can always be simplified by rewriting it as the sum of a whole number and a “proper” fraction:


whereas a “proper” fraction does not contain a whole number that can be pulled out of it:


A second reason is that “proper” and “improper” fractions will have opposite effects when used as factors.  When a number is multiplied by an “improper” fraction (whose absolute value is greater than one), the result will be larger than the original number:


When a number is multiplied by a “proper” fraction (whose absolute value is between zero and one), the result will be smaller (closer to zero) than the original number:


When an “improper” fraction is used as a factor in a multiplication problem, it grows the result.  When a “proper” fraction is used as a factor, it shrinks the result… which leads us to realize that there is more than one way to shrink a value: divide it by a number with an absolute value greater than one, or multiply it by a number with an absolute value between zero and one.

Undoing Multiplication

The discussion of proper and improper fractions may have led you to wonder if there is a way of undoing multiplication without resorting to division.  If a number has been multiplied by three, can the result be multiplied by some proper fraction  to produce the original number?  If so, what fraction will accomplish the goal?

5\cdot 3=15\\*~\\*15\cdot\frac{1}{3}=5

5\cdot 4=20\\*~\\*20\cdot\frac{1}{4}=5

The results above hopefully make sense intuitively, as our words describe what we are doing well.  Multiplying by one third undoes multiplying by three, as we are dividing the tripled quantity into three parts, then taking one of them… which will be the same quantity we started with.  Multiplying by one fourth undoes multiplying by four, as we are dividing the quadrupled quantity into four parts, then taking one of them, which will be the same quantity we had before multiplying by four.

Notice the relationship between three and one third, or four and one fourth. This relationship is so useful in mathematics that it has been given a name: a reciprocal.

To create the reciprocal of a number, divide one by the number.  The reciprocal of 5 is \frac{1}{5}, and the reciprocal of \frac{1}{5} is 5:


Division As Multiplication By The Reciprocal

Based on the examples of undoing multiplication given above, you can see that there are two ways of undoing multiplication by a factor: divide by the same factor, or multiply by the reciprocal of the factor.

Division is equivalent to multiplying by the reciprocal of the divisor:


This is a very useful property, as you will see in a minute.

Properties Of Division

Like subtraction, division is not associative:


and is not commutative:


so problems that involve division cannot be rearranged without changing the result.

To get around this problem, we rewrite division problems as multiplication by the reciprocal, just was we rewrote subtraction problems as addition of the negative.  The problem shown above becomes:


so that it can now be rearranged and/or regrouped easily using the commutative and/or associative properties of multiplication:

\dfrac{2}{10}\;\;=\;\;2\cdot\dfrac{1}{10}\;\;=\;\;\dfrac{1}{10}\cdot 2

A more complex division problem is usually much easier to work with when converted to multiplication by the reciprocal, which then allows you to rearrange it as you wish:

\dfrac{\left(\dfrac{5}{10}\right)}{2}\;\;=\;\;\dfrac{5}{10}\cdot\dfrac{1}{2}\;\;=\;\;5\cdot\dfrac{1}{10}\cdot\dfrac{1}{2}\;\;=\\*~\\*~\\*\;\;\left[\dfrac{1}{2}\cdot\dfrac{1}{10}\right]\cdot 5\;\;=\;\;\dfrac{1}{2}\cdot\left [\dfrac{1}{10}\cdot5\right]

Earlier posts in this series:
Algebra Intro 1: Numbers and Variables
Algebra Intro 2: Addition
Algebra Intro 3: Subtraction
Algebra Intro 4: Negative Numbers, Zero, Absolute Values, and Opposites
Algebra Intro 5: Addition, Subtraction, and Terms
Algebra Intro 6: Multiplication
Algebra Intro 7: Properties of Multiplication
Algebra Intro 8: Division 

Posts that continue this series:
Algebra Intro 10: Fractions and Multiplication
Algebra Intro 11: Dividing Fractions, Equivalent Fractions
Algebra Intro 12: Adding and Subtracting Fractions

By Whit Ford

Math tutor since 1992. Former math teacher, product manager, software developer, research analyst, etc.

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