It’s Professor Dave, I wanna tell you about the types of numbers.
We have already begun to look at some different types of numbers, but it is time to do a survey
of all the types of numbers that exist, and make sure we understand their characteristics.
To do this, we can look at the taxonomy of numbers just like we would look at the taxonomy
of biological organisms or fundamental particles.
We start at the top with all numbers.
Any number you can possibly conceive of, they are all here somewhere.
But some numbers are real, while some numbers are imaginary.
Real numbers are probably still all the ones you were thinking of, so let’s define imaginary
numbers.
Remember when we said that we can’t take the square root of a negative number?
In a certain sense you can’t, because there is no number that when squared will produce
a negative number.
However, in another sense, we can take the square root of a negative number if we admit
that the result is imaginary.
While the square root of one is simply one, we will say that the square root of negative
one is I.
This is a lower case I, always in italics, and it is the fundamental imaginary number.
The square root of negative four would be two I, because negative four is equal to four
times negative one, so this radical could be split up into root four times root negative
one, which is therefore two times I, or two I.
That covers imaginary numbers, so let’s go back to the real numbers.
Within the category of real numbers, there are two subcategories, rational and irrational.
Rational numbers are defined as any number that can be expressed as the ratio of two
integers.
Three halves, ninety-three six hundred and fourths, you just make a fraction, put any
integer on top and any integer on the bottom, and you’ve got a rational number.
The number on the bottom can also be one, so all the integers themselves qualify as
rational numbers.
The only number that can’t go on the bottom is zero, as that will yield a value that is
undefined.
Now, what are these irrational numbers?
Given the definition of a rational number, clearly an irrational number is one that can’t
be expressed as the ratio of two integers.
But what could these numbers be?
Irrational numbers are ones that have non-repeating decimals that are nonterminating, meaning
they continue forever with no pattern.
The square root of two is such a number.
If we plug this into a calculator, it will tell us that the answer is one point four
one four two, et cetera et cetera, but no matter how many digits your calculator displays,
it’s not giving you the full story.
This number extends forever, which means we can never list the square root of two as a
decimal and be infinitely accurate, the way we can with three halves.
Three halves, or 1.5, is exactly three over exactly two.
If we try to approximate root two this way, we can get very close, with something like
six hundred sixty five thousand eight hundred fifty seven over four hundred seventy thousand
eight hundred thirty two, but it’s not exactly right, and when we square this number, we
get something just ever so slightly larger than two.
No matter how precise you try to get with this ratio, you will never quite get to the
true value.
Now you might be wondering, if we can’t write out this number no matter what we do,
doesn’t that make this number imaginary?
The answer is no.
Nature is very clear in communicating to us that numbers like root two do indeed exist.
If you make a right triangle that has legs with a length of one, the third side will
have a length of root two.
This triangle exists, so root two exists.
It is a real number, it is simply irrational.
When it was discovered by the Ancient Greeks that irrational numbers must exist, all hell
broke loose, as it threatened the mystical notion that the world of numbers must obey
a divine perfection.
Other irrational numbers include pi, which is the ratio of the circumference of a circle
to its diameter.
We all probably know the first few digits of pi, which are three point one four one
five nine, yadda yadda yadda.
Again, this number never ends, with no pattern whatsoever in the digits.
And it’s not for lack of trying, mathematicians have checked.
With computers we can calculate pi out to literally trillions of decimal places, and
the numbers are random throughout.
Once again, even though pi is irrational, it is a real number, because circles are real,
and all circles have a circumference.
But we will get to triangles and circles when we start looking at geometry right after this.
Let’s quickly mention that there are numbers with nonterminating decimals that are actually
rational.
Take for example one third.
If we convert this to a decimal, we get zero point three, with the three repeating forever.
Although the threes go forever, this is a pattern, it’s three, then three, then three,
and so forth.
When decimals extend forever but in a pattern like this, they are actually rational, because
they can indeed be expressed as the ratio of two integers, in this case one over three.
Another example would be one ninth, or zero point one repeating.
That’s what this little bar means, whatever is under the bar repeats forever, in this
case just the number one.
Two ninths is equal to zero point two repeating, and this continues all the way up to nine
ninths.
That would be zero point nine repeating, and this is absolutely equivalent to one.
To prove this, we can rationalize that if the nines extend forever, then there cannot
be any number in between this number and one, and they therefore must be the same number.
There are other numbers with nonterminating decimals in a pattern of two or three repeating
digits, or more.
These are all rational numbers that are simply the decimal versions of some fraction.
If we want to, we can subdivide rational numbers further to describe types of numbers we already
know about, like fractions and integers.
Fractions are all the numbers that happen in between the integers, and there are infinitely
many of them in between each integer.
Within integers, we can describe whole numbers, which are all the integers starting from zero
and going in the positive direction, and natural numbers, which are all the integers starting
from one, so the same as whole numbers just without the zero.
And that’s pretty much all we need.
After all this, we should be able to describe the difference between real and imaginary
numbers, and within the real numbers, the difference between rational and irrational
numbers.
To make sure that’s the case, let’s check comprehension.