what is real value & what is real number??

REAL VALUE:
Nominal value is the value of anything expressed in money of the day, versus real value which removes the effect of inflation.

Nominal numbers - such as nominal wages, nominal interest rates, or nominal gross domestic product - refer to amounts that are paid or earned in money terms. A paycheck shows money wage and a car loan agreement indicates the nominal interest rate. Nominal GDP refers to the amount of money spent to buy the production of a country.

Real numbers - such as real wages, real interest rates, or real gross domestic product - are corrected for the effects of inflation. They indicate the value of these numbers in terms of the purchasing power of wages, interest, or total production. That is, they calculate how many goods and services a wage, an interest payment, or total domestic income will buy.

The nominal value is measured in the currency unit at that date. The real value is measured in stable purchasing power units. The reference unit can be the currency at the beginning of the period, the end, or the middle of the period. An example of reference units is constant dollars.

The measure of inflation that is removed to normalize the purchasing power unit can vary.

Real Gross Domestic Product (GDP) has had the GDP Deflator removed.
Real interest rates have had the Consumer Price Index (CPI) subtracted.
The expected real interest rate is the nominal interest rate minus the inflation rate expected over the term of the loan.
The realized (ex post) real interest rate has the actual inflation rate subtracted from the nominal interest rate.
The terminology of classical economics used by Adam Smith used a unit of labour as the purchasing power unit.
There are other measures of inflation used in other situations.
Nominal values are the default in most real life situations. Taxes are applied to nominal income, mortgages are written with nominal interest rates, wage agreements are written with nominal rate increases. These nominal rates are likely to have an inflation presumption built in, and only time will tell if the presumption was correct.

REAL NUMBER

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In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers. A real number may be thought of as any point on an infinitely long number line.

The discovery of more rigorous definitions of the real numbers was one of the most important developments of 19th century mathematics. Definitions in use today include equivalence classes of Cauchy sequences of rational numbers; Dedekind cuts; a more sophisticated version of "decimal representation"; and an axiomatic definition of the real numbers as the unique complete Archimedean ordered field. These definitions are all described in detail below.

The term "real number" is a retronym coined in response to "imaginary number".

First of all, I will try to define real number.

Historically, it is easy for human beings to understand what natural numbers are. They are just a tool for counting. After some cultural improvement, only natural numbers are not enough for human beings' daily usage, and they developed a sense of directed numbers, i.e. sense of positivity and negativity, and later they learnt about zero. At this stage the development of integers are complete.

Some thousand years later, ancient cizilizations raised and notably Greek started to study the ratio between two known length constructed by unity. This is the start of fractions, or rational numbers. The Greek thought that the world was so well-organized that everything was constructed by numbers, or ratio between numbers (by the way, the numbers they were talking about were integers indeed), but one of their hero and philosopher Pythagoras nearly destroyed all the things. Pythagoras is renowned now with the famous Pythagoras Theorem stating that the sum of two smaller squares constructed on the shorter sides of a right-angled triangle is equal to the largest square constructed on the longest side (hypotenuse). When an isoceles triangle with shorter sides are both 1, Pythagoras found that the side of the largest triangle cannot be any number in their sense, because it cannot be expressed as a ratio of two known lengths constructed by unity. They started to found that there were even something besides the ratio (rational number), and this was irrational number.

With rational numbers and irrational numbers together combined to real number.

For real-value, it is usually a value followed by a function which gives a real number as the function value. There is an extended meaning that if a function gives something else but a part of which is real number, the real value of this function means just the value of the real number part.