what are the importance of set R (real numbers ) in mathematics ?

Answer 1

Real numbers represent the set of objects satisfying the axioms of algebra (like high school algebra) which is really (really!) a special case of a "Field" which is used extensively in almost every type of technical human endeavor from rocket science to taxes and P-Chem to plumbing!

Answer 2

It is large enough (has enough numbers in it) to perform most relevant calculations.

First of all, it contains the counting numbers 1, 2, 3, 4, etc.
When you add, subtract, divide or multiply them, the result is always a real number. The real numbers are "closed" under this operation. Indeed, this makes it a field.

But that is not all. The set of rational numbers, Q, is also a field and is (much) smaller.

The real numbers are also topologically closed. When you make a series of numbers that comes infinitely close to each other (a Cauchy sequence), there is a limit which is also a real number. For instance, the infinite sum

1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 -+

has a limit, namely pi/4, which is not a rational number but is part of the real numbers.

These properties are the ones we need most of the time, and the real numbers have them all. For other applications we also need algebraic closure, and then the complex numbers are needed.

Answer 3

It's important, first of all, to know what numbers you will be working with.

Generally, when you look at a line, like y = 3x + 5, you don't tend to say that you are using all real numbers, R.

Sometimes it doesn't make sense to use all real numbers and instead you might use a subset of R. Like if I wanted to measure heights, you wouldn't say that you were 5' 3.1234235 inches. You would round to the nearest inch. So there, you would use Natural numbers (or counting numbers.)

For temperatures, you may have a range of values, like -25 F to 120 F. You may never experience a temperature out of that range so it's not worth including it in your graph. Then you define your range as {R, where temperature is between -25 and 120}

There are numbers that are not real! So it's important to know all the types of Real numbers out there!

I hope this helped!

Answer 4

What's is the importance of an engine to your car? My point about real numbers is they are the fundamentals upon which mathematics is built. Maybe a better analogy is what is the importance of a foundation to a building? And I don't think Gauss and Möbius had small minds for working with imaginary numbers...

Answer 5

All the history of humanity has been made usig those numbers.

Imaginaries numbers as a concept are uses only such late as in 1685 (John Wallis (1616-1703))

Answer 6

Actually, and number that is not a real number is useless. They are invented and used by small minds that cannot comprehend something, so they invent an explanation for it.

Answer 7

Because fake numbers don't work.