Hello. I'm Professor Von Schmohawk and welcome to Why U.
In our lectures so far
we have seen how numbers evolved from the primitive concept of natural counting numbers
to whole numbers with the addition of the number zero
integers, with the addition of negative numbers
and then rational numbers.
Since rational numbers are closed under all four arithmetic operations
addition
subtraction
multiplication
and division
you might think that rational numbers finally represent a complete collection of numbers.
We can add, subtract, multiply, and divide
and never have a problem representing the result.
This is what the Greeks thought as well
that is, until they discovered
an inconvenient truth
that there are certain types of quantities that cannot be represented by rational numbers.
Nobody knows exactly who made this disturbing discovery.
It may have been the great Greek mathematician Pythagoras.
But legend has it, it was one of his students
who was subsequently killed
for upsetting the apple cart with this discovery.
Numbers were obviously a sensitive subject for the Pythagoreans.
The proof of the existence of irrational numbers may have gone something like this.
If rational numbers can represent any quantity
then they should be able to represent square roots.
Now the definition of rational numbers says that
any rational number can be represented as the ratio of two integers
in other words
one integer which we will call p
divided by another which we will call q.
p over q is obviously a fraction
and like any fraction we can write it in its simplest form
so that the integers p and q have no common factors.
We also know that since the square root of two is not an integer
the denominator q is some number other than one.
Now, since the "square root of two" and the quantity "p over q" are equal
then we can square them both and they will still be equal.
The "square root of two" squared is two
and the quantity "p over q" squared is "p over q" times "p over q"
or "p times p" over "q times q".
For this fraction to be equal to two
we must somehow be able to reduce it to two over one.
But we already saw that q was not one.
So the only other way this fraction could be equal to two over one
is to cancel out common factors in the numerator and the denominator.
But we also said that p and q have no common factors.
Therefore, there are no factors in the numerator that are the same as any factor in the denominator.
So we can't cancel any factors, and there is no way this fraction can be equal to two!
Therefore, this is an impossible situation.
Our original assumption
that the square root of two can be represented by the ratio of two integers
must be wrong.
The square root of two must be some value which is not a rational number.
A number which is not rational is called an "irrational number".
This same argument can work for other square roots.
For example, we can start with the square root of three
and try and represent it as the ratio of two integers with no common factors.
Squaring both sides
we get three equal to "p times p" over "q times q"
and since p and q have no common factors
this fraction can't be reduced to the integer three.
Therefore, the square root of three is irrational.
This also works for roots other than square roots.
For example, we can set the cube root of two equal to the same ratio with no common factors.
Then cubing both sides
we get two equal to "p times p times p" over "q times q times q"
and since p and q have no common factors the fraction can't be reduced to an integer
and so the cube root of two is also irrational.
We saw in the chapter on Converting Fractions to Decimal Numbers
that rational numbers can be represented as decimal numbers which either repeat or terminate.
A terminating decimal number has a finite number of digits after the decimal point
while a repeating decimal number ends with an infinitely repeating series of digits.
The repeating sequence of digits can be as long as we like
but the number of digits which repeat is always finite.
In other words, all repeating decimal numbers
contain some finite number of digits which repeat infinitely many times.
However, when we try and represent an irrational number as a decimal number
we find that the number will always contain an infinite sequence of non-repeating digits.
No matter how many digits we write after the decimal point, the digits will never repeat.
Therefore, we can never perfectly represent an irrational number
either as a decimal number or as a fraction.
With the invention of calculus
mathematicians were able to prove that numbers like pi and e are also irrational.
In fact, there are an infinite number of irrational numbers.
In the next lecture we will see how mathematicians
eventually combined the irrational numbers with the rational numbers
to form the real numbers.