Explain the concept of real numbers.

 

All the numbers on a number line that runs from negative infinity through zero to positive infinity are considered to be real numbers. The set of real numbers was evolved from the natural numbers used for counting, rather than being constructed arbitrarily. 

Numerous discrepancies exist in the system of natural numbers, and as computations became more intricate, the number system evolved to overcome these issues. Real numbers have minimal exceptions or restrictions compared to earlier iterations of the number system, which made computations more reliable and consistent.

Closure and Natural Numbers.

A set of numbers may be said to be closed if all permitted computations are carried out on numbers that are members of the set, and the results are likewise numbers that are members of the set. The set is supposedly finished.

Natural numbers are the counting numbers, such as 1, 2, 3, etc.; their set is unbounded. As soon as natural numbers were utilised in trade, two issues appeared. While the natural numbers could count actual things like cows, there was no natural number for the outcome if a farmer possessed five cows and sold five cows. Number 31 is natural real numbers and factos of 31 are 1 and 31 are also natural numbers.

To solve this issue, early number systems created a word for zero relatively fast. The system of whole numbers, which is composed of the natural numbers plus zero, is the outcome.

Subtraction was also a factor in the second issue. The farmer could not sell more cows than he possessed as long as numbers counted actual items like cows. However, as numbers became abstract, greater numbers were subtracted from smaller ones to get results that did not follow the rules of whole numbers. 

Integers, which are whole numbers plus negative natural numbers, were created as a consequence. There was now a whole number line in the number system, although it only included integers.

Logic Numbers.

Calculations in a closed number system should provide results for operations like addition and multiplication as well as its inverse operations, subtraction and division, from inside the number system. For addition, subtraction, and multiplication but not division, the integer system is closed. The outcome of dividing one integer by another is not necessarily another integer.

A fraction is produced when a small integer is divided by a bigger one. These fractions were included as rational numbers in the number system. Any number that can be written as the ratio of two integers is referred to be a rational number. A rational number may be used to represent any decimal number. 2.864 is equal to 2864/1000 and 0.89632 is equal to 89632/100,000. Now, the number line seemed to be finished.

Unrealistic Numbers

On the number line, there exist certain numbers that cannot be written as a fraction of integers. The hypotenuse to side ratio of a right-angled triangle is one. The hypotenuse of a right-angled triangle is equal to the square root of two if two of its sides are 1 and 1. The decimal representation of the square root of two is limitless and non-repeating.

All actual numbers that defy reason fall under the category of such "irrational" numbers. Since every other real number that is not rational is included in the definition of irrational, this definition completes the number line for all real numbers.

Infinity

Infinity itself is not a real number; rather, it is a notion of the number system that defines it as being an amount bigger than any number, even if it is asserted that the real number line extends from negative to positive infinity. 

As x approaches 0, the mathematical solution to 1/x is infinity, although division by zero is not specified. Because infinity defies the rules of arithmetic, if it were a number, it would produce paradoxes. For instance, infinite plus one remains infinite.

Unreal Numbers

Except for division by zero, which is undefined, the set of real numbers is closed for addition, subtraction, multiplication, and division. For at least one further action, the set is not closed.

According to the laws of multiplication in the set of real numbers, multiplying a negative number by a positive number results in a negative number, whereas multiplying a positive number by a negative number results in a positive number. 

In other words, both positive and negative numbers produce a positive number when multiplied by themselves in the specific situation. The square root of a positive integer, which yields both a positive and a negative result, is the inverse of this exceptional instance. There is no solution in the set of real numbers for the square root of a negative number.

The idea of a set of fictitious numbers attempts to solve the problem of negative square roots in real numbers. All imaginary numbers are multiples of I which is the square root of -1. The set of complex numbers must include both all real and all imaginary numbers in order for number theory to be complete. Real and imaginary numbers may be represented as horizontal and vertical number lines, with zero marking the intersection of the two. Complex numbers are points having both real and imaginary components in the plane formed by the two number lines.