Complex Numbers vs Real Numbers
Real Numbers and Complex Numbers are two terminologies often used in Number Theory. From the long history of evolving numbers, one must say these two play a huge role. As it suggests, ‘Real Numbers’ mean the numbers which are ‘Real’. In the meantime, ‘Complex Numbers’ as the name refers a heterogeneous mix.
From the history, our forefathers used numbers to count the livestock to keep them in check. Those numbers were ‘Natural’ since all of them are simply countable. Then the special ‘0’ and the ‘Negative’ numbers were found. Later, ‘Decimal Numbers’ (2.3, 3.15) and numbers like 5⁄3 (‘Rational Numbers’) were also invented. The main difference between aforesaid two different types of decimals is that one ends with a definite value (2.3 Finite Decimal) while the other repeats according to a sequence, which in the above case 1.666… Thereafter an interesting phenomenon came into picture, that of course the ‘Irrational Number’. Numbers like√3 are examples for such ‘Irrational Number’. Eventually intellectuals found another set of numbers which are denoted in symbols as well. A perfect example for that is the most familiar face of π, and represented by the value 3.1415926535…, a ‘Transcendental Number’.
All the above mentioned categories of numbers embrace under the name of ‘Real Numbers’. In other words, Real numbers are the numbers which could be depicted in an infinite line or real line where all the numbers are represented by points. Integers are equally spaced. Even the Transcendental Numbers are also pointed exactly by increasing the number of decimals. The last digit of a decimal decides that respect to which tenth of an interval that number belongs to.
Now if we turn the tables and look the insight of ‘Complex Numbers’ which can be easily identified as a combination of ‘Real Numbers’ and ‘Imaginary Numbers’. Complex extends the idea of a one dimensional into two dimensional ‘Complex Plane’ comprising ‘Real Number’ on the horizontal plane and ‘Imaginary Number’ on vertical plane. Here if you don’t have the glimpse of ‘Imaginary Number’, simply imagine√(-1) and what guess what would be the solution? Ultimately the famous Italian mathematician found it and denoted it ‘ὶ’.
So in detailed view, ‘Complex Numbers’ consist of ‘Real Numbers’ as well as the ‘Imaginary Numbers’, whereas ‘Real Numbers’ are all which lies in the infinite line. This gives the idea ‘Complex’ stands out and holds a huge set of numbers than ‘Real’. Eventually all the ‘Real Numbers’ can be derived from ‘Complex Numbers’ by having ‘Imaginary Numbers’ Null.
Example:
1. 5+ 9ὶ: Complex Number
2. 7: Real Number, However 7 can be represented as 7+ 0ὶ as well.