All the elements former commuted of group under addition, and if you throw out zero, the remaining elements form a community group under multiplication.

This is the more formal definition.

Let's now look at some examples of fields.

Theo integers are not a field.

We denote the integers by a double stroke C, also known as a blackboard.

Bold.

See, we have two operations, addition and multiplication.

The distributive property holds, and under addition, the integers even form a commuted of group, also known as an a 1,000,000,000 group.

But we run into trouble when we look at multiplication.

The integers don't have in verses under multiplication.

The multiplication of inverse of two is 1/2 and 1/2 is not an integer.

The multiplication of inverse of three is 1/3 also not an integer.

And so, uh, but if we add all the fractions to the set of integers, we get a field the rational numbers we call fractions rational numbers because they're all ratios of integers.

The set of all fractions is denoted by acute for quotient.

The additive inverse of a fraction say, for example, 3/5 is negative 3/5 which is another fraction the multiplication of in versus 5/3.

Again, a fraction.

So after you toss out zero the set of fractions all have multiplication.

Vin vs.

The rational numbers are a field.

We can take the set of rational numbers and turn them into a larger field.

Take, for example, the square root of two.

We've known for thousands of years that this is not a rational number.

Is there a field that contains this number?

Indeed there is.

What if we took the square root of two and toss it into the set of rational numbers?

We then add, subtract, multiply and divide all these numbers together.

And we do this over and over until we have a new, larger set of numbers, which is a field containing the square root of two.

Here is a notation for this field.

We call it an extension.

Field of the rational numbers of very good exercise is to convince yourself that every number in this field can be written in this form.

Our previous example opened up a whole new world of fields.

For us.

You could extend the rational numbers by adding the cube root of three or I, the square root of negative one.

There are an infinite number of extension fields of the rational numbers.

In fact, for any polynomial equation, you can take a solution to this equation.

Add that to the rational numbers and get an extension field.

We call this an algebraic extension, and you study these a locked and number theory and calculus.

You learn about conversion sequences, a series of numbers which get closer and closer to some value.

We call this value the limit of the sequence.

This idea will lead us to another field.

If you start off with the rational numbers and then look at all the convergent sequences, take their limits and throw them into the set of rational numbers.

You will end up with the real numbers, a very large and well known field.

The notation for the real numbers isn't are are for real.

And because we've added all the limits and filled in all the gaps, we say the real numbers are complete.

There are still some numbers missing from the field of real numbers.

I, for example, the square root of negative one.

If you add I to the real numbers than add, subtract, multiply and divide everything.

Repeatedly, you end up with the complex numbers.

We denote the complex numbers with the sea.

You may be thinking at this point, can we go bigger?

Can we make bigger fields by finding new numbers, either by taking limits of sequences or solutions to polynomial equations?

The answer is no.

The field of complex numbers is quite large.

Any sequence of complex numbers, which converges, will converge to another complex number.

It is complete, and any polynomial equation can be solved completely.

Using complex numbers, we say the complex numbers are algebraic.

We closed.

So is this the end of the line?

Can we not make any more fields?

Actually, we can, but we have to think a bit differently.

What if, instead, of adding a new number, we added a variable X.

Once we add X to the mix, we get the set of rational functions fractions where the numerator and denominator are both polynomial sze.

Believe it or not, this is a field to the field of rational functions over the complex numbers.

We can even make this larger by adding a second variable.

Why or third Z?

I think we should stop there for now.

As you can see, there are a lot of different fields in this world.

One of the goals and abstract algebra is to classify all these fields and investigate their properties.

And while we focus today on infinite fields, there even field with only a finite number of elements thes air called finite fields, and we'll talk about those in a separate video.