Real Numbers

a real number is any number that is not imaginary

an imaginary number is a multiple of sqrt(-1)

Re: Real Numbers

All real numbers are rational numbers. - false

(A real number is defined as a limit of a sequence of rational numbers. - true)

The set of all rational and irrational numbers

http://www.maths.abdn.ac.uk/~igc/tch/ma2001/notes/node8.html

g_day, your link does not define the rational nor the irrational numbers..

Re: Real Numbers


You must only know that:

N c Z c Q (where "N c Z" means "N is a subset of Z").
Q u I =R (where "u" means union, joint and I is the set of irrational numbers)

Therefore, the statement is false. It's true, however, that "All rational numbers are real numbers". Numbers that are not real are complex (4+i, and so on).

I hope you have understood it.

False

Real numbers include all numbers that are not imaginary. This includes the Irrational numbers, such as (pi, phi, √2, √3, √5, √7) and the rational (numbers that can be expressed as ratios of integers).

Rap

that is what we leant last year

You were half right: All rational numbers are real numbers.

Squares are rectangles too.

And listen to Stuh - because you can have both rational and irrational imaginary lines.

And don't forget the infinity of points on a line is a larger infinity than the infinity of rational numbers.

I wonder if the infinity of imaginary numbers is as large or larger than the infinity of real numbers.

See, now you've got me going. Is George Gamov in the room?

If I said "imaginary lines" I meant imaginary numbers - both rational and irrational. Pi is the most appealing of irrational numbers or at least the most appetizing. Would you care for a slice?

"I wonder if the infinity of imaginary numbers is as large or larger than the infinity of real numbers."

cardinality of C = cardinality of RxR = cardinality of R

The amazing thing is that between two rational numbers there is an irrational number and vice versa, but the cardinality of irrational numbers is by far bigger than the cardinality of rational numbers.

That aside, I think it's time someone proved to our little child that not all real numbers are rational. Let's assume sqrt(2) is rational (actually one should also prove that sqrt(2) is real, but high school students assume it is).
sqrt(2)=p/q such that p and q are coprime (there is no prime that divides both p and q because then we can illuminate it).
Lets multiply each side by itself.
2=p^2/q^2
=> 2*q^2=p^2
Thus 2 divides p^2. That also means that 2 divides p. Lets write p as 2*k.
2*q^2=4*k^2 => q^2=2*k^2
Thus 2 divides q^2. That also means that 2 divides q. But we assumed that p and q are coprime... Contradiction. That's why sqrt(2) is irrational.

There is a proof I like more using abstract algebra, but it uses concepts that are a bit more difficult to understand.