What is a real number?

Math Adda                                      

Real numbers

What is a real number?

A real number is distinguished from an imaginary number.
It is a number we require for measuring rather than counting. Hence a real number is any rational or irrational number that we might name. They are the numbers we expect to find on the number line.

(An actual measurement can result only in a rational number.
 An irrational number can result only from a theoretical  calculation; examples are the Pythagorean theorem, and solving
 an equation such as x³ = 5.
 Any serious theory of measurement must address the question:  Which irrational numbers are theoretically possible? Which ones  could be actually predictive of a measurement?)

Problem 1.   We have categorized numbers as real, rational, irrational, andinteger.  Name all the categories to which each of the following belongs.

3  Real, rational, integer.		−3   Real, rational, integer.
−½   Real, rational.		Real, irrational.
5¾   Real, rational.		− 11/2   Real, rational.
1.732   Real, rational.		6.920920920. . .   Real, rational.
6.9205729744. . .   Real. And let us assume that it is irrational, that is, that the digits do not repeat. Moreover, we must assume that there is an effective procedure for computing each next digit. For if there were not, then we would not know which number we are computing. And that symbol would not refer to any "number"!
6.9205729744    Real, rational. Every exact decimal is rational.

7.  What is a real variable?

A variable is a symbol that takes on values. A value is a number.
A real variable takes on values that are real numbers.

Calculus is the study of functions of a real variable.

Problem 2.   Let x be a real variable, and let 3 < x < 4.  Name five values that x might have.