How to Understand the Concept of Real and Complex Numbers (Algebra)

Step-by-Step Guide

1. 1

   Step 1: Study the Real numbers also called the set of "R = Reals".

   Division by zero is undefined.
   
   Rational numbers = all the naturals, wholes, common fractions, integers, and some other non-common fractional expressions, or decimals and roots that all have a terminating decimal or repeating decimal equivalent form.
   
   Fractions, equivalent decimals and percentages are three examples of rational numbers:
   Fractions = p/q (p and q are integers, but q is not zero).
   
   Also, terminating decimals and repeating decimals are other examples of rationals, like:
   Terminating decimals are such as: 1/4 = .25 = 25/100 = 25%, and 3/4 = .75 = 75/100 = 75%, and 5/8 = .625 = 625/1000 =
   62.5/100 =
   62.5% .
   
   Repeating decimal like 1/3 = .333... = 33 1/3% or ~
   33.333%, and also 2/3 = .666... = 66 2/3% or ~
   66.66% (often rounded to
   66.7% = .667 for convenience). and in 3/13 =
   0.230769230769230769... notice that "230769" repeats, which indicates that 3/13 is rational (and, of course, any rational divided by a rational could be shown to have a rational quotient). ,, Irrationals, stated as decimals neither terminate nor repeat.
   
   They can not be stated as a ratio such as a fraction (p/q) that would terminate or repeat in its decimal equivalent.
   
   The number π, e and sqrt(2), of 3, or of 5,..., and the 3rd root(4) are some examples of irrationals. ,, Imaginary numbers are based on i = (sq root of
   -1) = i1 or 1i.
   
   The i is the symbol for imaginary numbers, such that: (a.) 1i X 1i =
   -1, but since (b.)
   -1 X
   -1 = +1, just as (c.) 1 X 1 = +1 Example: the sqrt(-9) = 3i is called "imaginary 3"
   
   or "3 imaginary" and is not Real.
   
   Check: i3 X i3, or 3i X 3i =
   -9.
2. 2

   Step 2: Build understanding of the system of Reals by knowing the following subsets of the set of rational numbers

   Example:
   The sqrt(-8) is imaginary = " i(2nd root(8)) "
   
   Example: 3rd root of (-8) =
   -2 is Real.
   
   Check:
   -2 X
   -2 X
   -2 =
   -8. , bi is the imaginary part of that value. ax is Real as seen here. , Here is how Reals are easily defined as y = ax based on y = ax + bi.
   
   Show that subset y = ax + 0 is where the (optional) imaginary part bi is not present (not relevant).
   
   Why not
   -- because we have b = 0 and bi = 0i = 0, so then we call that the values of a subset R, such that y = ax makes up all of the Real numbers.
3. 3

   Step 3: including: Naturals

4. 4

   Step 4: N = {1

5. 5

   Step 5: (the counting numbers) Wholes

6. 6

   Step 6: W = {0

7. 7

   Step 7: (Naturals and zero) Integers

8. 8

   Step 8: P = {...

9. 9

   Step 9: ...} Common Fractions

10. 10

    Step 10: p/q which means the quotient of p divided by q

11. 11

    Step 11: where q is not zero

12. 12

    Step 12: p and q are natural numbers (quotient - the solution of a division).

13. 13

    Step 13: Draw a line (called a bar) "above the part of the number that repeats" in a repeating decimal: this is when handwriting

14. 14

    Step 14: to indicate the repeating part of that rational number.

15. 15

    Step 15: Study "Irrational" numbers.

16. 16

    Step 16: Learn that the Real numbers are all of the above: made up of the rationals and the irrationals

17. 17

    Step 17: and does not include the imaginary numbers.

18. 18

    Step 18: Study the imaginary numbers.

19. 19

    Step 19: Consider this rule: Even roots of negative real numbers are imaginary.

20. 20

    Step 20: Consider another rule: Odd roots of negative reals are negative Reals.

21. 21

    Step 21: Define the complex numbers are of form y = ax + bi.

22. 22

    Step 22: Define Reals are a subset of the set of complex numbers.

Detailed Guide

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