Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the previous lectures, we saw that
although multiplying natural numbers always produces another natural number,
the result of division between natural numbers
cannot always be represented by a natural number.
For example, if we divide one by three,
we must split one unit length
into three equal pieces.
Unfortunately, we don’t have any way to represent increments smaller than one
using natural numbers or integers.
To solve this problem, ancient people as long ago as the Bronze Age
came up with the concept of a fraction.
Historically, the earliest fractions were "unit fractions".
A unit fraction is a very simple type of fraction.
Unit fractions are written as one over any natural number called the denominator.
We will represent this number with the letter q.
Examples of unit fractions are
one-half, one-third, one-fourth, and so on.
The idea of a unit fraction is simple.
If a unit length is divided into some number of equal pieces,
a unit fraction represents the length of one piece.
The denominator tells us how many of these pieces fit into one unit length.
Unit fractions were easy for ancient people to understand
since they, in effect, undid multiplication.
For instance, let’s say that you had a plot of land
and your two brothers, Amin and Hakim, also have equal size plots.
Your bothers decide to give you their land and move to Mesopotamia.
You have now multiplied the size of your land holdings by three.
If the pharaoh then took most of your land to build a pyramid
but left you with one-third,
your land would then be the size of your original plot.
So keeping one-third of your land undid the operation of multiplying your land by three.
To get a better understanding about what unit fractions represent,
let’s look at unit fractions on the number line.
There are an infinite number of unit fractions.
Unfortunately, not all quantities between zero and one
can be represented by a unit fraction.
The Egyptians had a unique way to represent these quantities.
If they needed to represent something which was smaller than one,
but was not the exact size of a unit fraction,
they would add unit fractions until they achieved the desired amount.
However, for some reason which we do not understand today,
the ancient Egyptians had a rule that when adding unit fractions,
no two fractions which were added could be the same fraction.
For example, instead of writing that two-fifths of someone’s land was flooded,
they would have to write that one-third plus one-fifteenth of the land was flooded.
This complicated representation of fractional quantities became known as Egyptian fractions.
The Egyptians kept tables to convert various quantities between zero and one
to Egyptian fractions.
The Rhind Mathematical Papyrus from 1650 BC, contained such a table.
This table expressed various fractional quantities as sums of two, three, or four unit fractions.
Unit fractions were the only type of fractions used for thousands of years.
Not until the Middle Ages did people start using what were called "vulgar" fractions.
These fractions which today are called "common fractions" or just "fractions"
are composed of two integers which we will represent by the letters p and q.
The number on the top is called the "numerator" and the number on the bottom is the "denominator".
We can think of a common fraction
as being a unit fraction multiplied by the number in the numerator.
In other words, a common fraction is simply a multiple of a unit fraction.
As an example, let's take the fraction three-fifths.
Since the denominator is five,
the unit length on the number line is divided into five equal subdivisions.
The numerator specifies the number of those subdivisions in the fraction,
so the fraction three-fifths would represent three of those subdivisions.
So in any fraction, the denominator divides the unit length into equal parts,
and the numerator specifies the number of those subdivisions in the fraction.
In the previous lectures,
we have seen that integers are closed under addition, subtraction, and multiplication,
since the result of applying any of these operations to integers
always produces another integer.
However, when dividing integers, the result cannot always be represented by an integer.
For example, dividing the integer one by the integer three does not produce an integer.
Therefore, the integers are not closed under the operation of division.
To represent the result of division operations,
we will have to expand our number system to include fractions.
This collection of integers and fractions is called the "rational numbers".
Rational numbers are closed under addition, subtraction, multiplication, and division.
Rational numbers are defined as any number
whose value can be represented as a fraction p over q
where p and q are both integers.
The value of the rational number is the quantity obtained by dividing p by q.
We must add the caveat that q cannot be zero, since dividing anything by zero is meaningless.
Notice that p over q can not only represent any fraction,
but can also represent any integer if we let q equal one.
In that case, the fraction p over q represents the same quantity as the integer p.
Fractions can be thought of as ratios.
For instance, if two quantities have a ratio of two to one,
this means that the first quantity is twice as large as the second.
Likewise, a ratio of one to two
means that the first quantity is half as large as the second.
So the fraction one-half can be thought of as the ratio of the integers one and two.
In fact, this is where the term “rational” comes from.
“Rational number” means “a number which can be expressed as a ratio”.
With rational numbers, we finally have a number system
which can represent the result of any arithmetic operation.
In the next few lectures we will explore arithmetic operations
using various types of rational numbers and fractions.