What numbers aren't real?

Question

the other day in Algebra 1 we learned about real numbers and stuff

i was wondering

what numbers aren't real?

Answer 1

Numbers that aren't real are called 'imaginary' numbers.

For example, a number divided by 0 is an imaginary number because you can't divide a number by zero.

Another example is the square root of a negative.

You can not get the squareroot of -16 because two of the same numbers can't multiply itself to get -16.

If you do squareroot -16, it would be 4i (you pretend the negative isn't there then do the problem and add an i to the end.)

Answer 2

Well the real numbers belong to another group of numbers called complex numbers (sometimes called imaginary numbers).  

Have you ever tried calculating the square root of a negative number on your calculator?  If you haven't give it a try and you'll get an error message.  Now here is where complex numbers come in handy.  In mathematics we definite i = √(-1).  Now lets say we were trying to calculate √(-4), you obviously can't use your calculator because of the negative, so you'll have to use i = √(-1).

So write it like this:

√(-4)

= √[(-1)(4)]; breaking up -4 = (-1)(4)

= √(-1)√(4); by exponent rules

= 2i; by the definition of i.

Now see why it is useful?

Now the reason why I bring up this weird thing called 'i' is because a complex number is of the form:

z = a + bi, where a and b are real numbers and i = √(-1) .

'a' is called the real part and 'bi' is called the imaginary part for reasons you'd probably be able to guess.

Now if you notice that if we let b = 0 we get:

z = a, but a is a real number so this shows that the real numbers are subgroup of the complex numbers.  Therefore, real numbers are complex numbers with b = 0.  Whenever, b ≠ 0 we'll have a complex number which is not a real number.

Hope this helps!

Answer 3

Imaginary numbers are not real numbers, I'd rather call them complex numbers though, as they are just as much numbers as real numbers are.

An imaginary number takes the variable i, and is defined as i^2 = (-1).

(Dividing by zero is never defined as an imaginary number. It usually takes the name, infinity, but is sometimes denoted to be of indeterminate form.)