The last “arithmetic” operation introduced in school is usually division. While multiplication allows us to calculate the total needed for a group when a fixed quantity is required for each person, division allows us to determine how much each person will get when a fixed quantity is divided equally among all in a group. Division is the inverse of multiplication.
But how does one go about dividing? Should we use our knowledge of multiplication to intuitively back into what the answer has to be, or is there a more procedural way of solving a division problem?
Inverse Of Multiplication, Unscaling
Just as subtraction undoes addition, division undoes multiplication. If multiplication scales a quantity by a factor, division “unscales” it by the same factor – thereby returning it to its original value.
Multiplying four by three stretches the quantity “four” by a factor of “three”: the original quantity is made longer. Dividing this result by three undoes the stretching or scaling, and returns us to the starting value of four. If multiplying by a number stretches, dividing by that same number shrinks.
However, this approach does not (yet) help us calculate the result of division if we do not know the starting point that was used for the original multiplication.
Just as multiplication by an integer can be thought of as “repeated addition”, division by an integer can be thought of as “repeated subtraction”: twelve divided by three can be interpreted as “how many times can we subtract three from twelve before getting back to zero?”
This can work quite nicely for some problems, but runs into trouble for others. For example “eleven divided by five”: we can subtract five twice, but that leaves us at one – not zero. So the answer is more than two but less than three.
The Integers Are Not Enough
Just as subtraction pulled us out of the simpler world of “counting numbers” into the world of the “integers”, division forces us to expand our universe of numbers once again. The integers seemed to be all we needed for addition, subtraction, and multiplication: add, subtract, or multiply any two integers, and the result will always be an integer.
However, the integers can only represent the answers to some division problems. A division problem such as “eleven divided by five” has an answer that is greater than one integer, but smaller than the next one. We need to create some numbers that fit between integers.
Revisiting “eleven divided by five”, we can rephrase the answer as “two and one part out of five”. After subtracting two fives, we sought to subtract one more five, but could not do so as we only have one part of the necessary five left to subtract. Division introduces the need for numbers that lie between integers.
One way of describing a number between two integers is by using a number line. Take the distance between zero and one on the number line, then divide that distance into five equal parts. The quantity I need can be represented by the end of the second of those five parts: .
A rational number is one that can be represented as the ratio of two integers. Two parts out of five is an example of both a fraction (a quotient of two integers) and a rational number. All fractions are rational numbers (, ) however many rational numbers (such as 3, or 10) that can be expressed as fractions need not be written as fractions .
“One part out of five” represents a number that is between zero and one. It would take five of this number to add up to one. It is the number which, when subtracted five times from one, returns us to zero. It is the number we get when we divide one by five, and is more commonly referred to as a “fraction”. We usually refer to this number as a “fifth”.
If seventeen is divided by five we get “three and two fifths”, or three ones plus two fifths. Just as “three” means “three of the quantity one”, “two fifths” means “two of the quantity a fifth”.
Rational numbers can also represent whole numbers or integers: “six thirds” is a quantity that can be broken into two groups of “three thirds”. Since each group of three thirds adds up to one, six thirds is equivalent to the integer two.
In the elementary school I attended, division was usually represented using a division sign (a subtraction sign with one dot above and one dot below it):
However, when fractions were introduced, they were represented in a simpler way, using a horizontal line:
And then computers came along, and in programmers’ attempts to represent the arithmetic operations using the limited character set available on teletype keyboards, a slash was often used to represent division (as it was similar to the line used above):
As used above, all three of these notations mean exactly the same thing. However, there is one additional property of the “fraction bar”, or “vinculum” (the horizontal line used in the second example above). It can serve as a grouping symbol at the same time as it indicates division, whereas the other two symbols apply only to the terms adjacent to them, forcing parentheses to be used if we need to change the sequence dictated by the “order of operations”:
Division By Zero
And finally, there is a conundrum: what happens when something is divided by zero? Let’s use the following as an example:
Using the repeated subtraction model, this would require us to find how many times zero should be subtracted from five to return us to zero. If nothing is ever subtracted from five, we can never return to zero.
Using the inverse of multiplication model for the above, we could ask “what number, when multiplied by zero, produces five?” Since any number multiplied by zero will produce zero as the answer, there is no way to get a five as the result of a problem that involves multiplying by zero.
Dividing by zero does not make sense. The result of any problem that involves dividing by zero is therefore labelled as “undefined”. Mathematicians go to great lengths to avoid setting up problems in ways that could lead to dividing by zero. It is not always possible to do so, so you will occasionally see certain values excluded from problems in order to avoid dividing by zero.
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
Posts that continue this series:
Algebra Intro 9: Fractions, Reciprocals, and Properties of Division
Algebra Intro 10: Fractions and Multiplication
Algebra Intro 11: Dividing Fractions, Equivalent Fractions
Algebra Intro 12: Adding and Subtracting Fractions
Hi great site and articles, very informative.
“All fractions are rational numbers (, ) however some rational numbers are not fractions (3, 10).”
This I am confused about. I know that all integers are a subset of rational numbers, as all integers can be written as fractions or decimals. So how come 3 or 10 be non rational (ie irrational by definition), but integers (or natural numbers)?
Even though integers may not be written as fractions, they can be (3 = 3/1 =30/10 etc) we just take the simplest form and ignore the denominator of 1.
Do you mean “pi” and thus irrational numbers like sqrt 2 etc, then I would agree.
If am mistaken or have some misconception, I would love to hear an explanation
If a fraction is a number that *is* written as a ratio of two numbers, then 3 is not written as a fraction, while 3/1 is written as a fraction. Both versions represent the same quantity, as would 3.0, 6/2, etc. Therefore, “3” is not a fraction, while “3/1” and “6/2” are, yet they all represent the same quantity.
All integers are rational numbers: they *can* be, but do not have to be, expressed exactly as a ratio of two integers. 3 is a rational number because a completely equivalent quantity *can* be exactly expressed as a ratio of two integers: 3/1, 6/2, etc.
Irrational numbers, such a pi, cannot – no matter how hard we try – be represented exactly as a ratio of two integers (approximations may exist, but exact versions do not). 22/7 is close to the value of pi, but is not exactly pi. Since there is no exact representation of pi as a ratio of two integers, it is an irrational number.
Based on your question, I will change my wording in the blog post you quoted to end “many rational numbers that can be expressed as fractions need not be written as fractions (3, 10).”
Thank you for your close reading and question!