I can’t understand the reason why a number raised to zero will give a 1 always? Help needed.
Kuldeep Singh
2

After much thought, I have realized that x^0 = 1 is a joke on singles. It basically means you stay single (1) if you don’t multiply (x^0). :D

Proof:

  1. x^k = y [Read: You would have to multiply x successively k times to get y; k is called exponent]
  2. Inversely, k = log_x(y) [Read: You can divide y successively by x at most k times; the same k is called logarithm]
  3. 0 = log_x(1) [Read: At most how many times can you divide 1 successively by x? 0 because you can’t—1 is indivisible.]
    Therefore, raising x to the 0th power will give you 1 rewriting (3) according to (1) and (2) as:
  4. x^0 = 1 [Read: How many times can I successively multiply x to get 1? 0. You stay single if you don’t multiply (multi = more than 1; multiply = become more than 1)—cheeky :D but that’s what it means.]

QED.

Another Proof:

  1. x^m/x^n = x^(m-n) [exponentiation rules]
  2. a/a = 1 [rules of arithmetic]
  3. When m == n, 
    1 = x^m/x^m = x^(m-m) = x^0 [by (1) and (2)]

QED.

Background:

A logarithm counts successive divisions — it is the number of times you would have to divide a real number Y by another real number X until you cannot divide further (you reach 1; 1 is “indivisible”). Therefore, a logarithm basically answers the question: “how many times can I successively divide a number until it can no longer be divided?” Exponent is another term for logarithm, but goes in the other directionit counts successive multiplications. Logarithm is about reduction while exponentiation is about growth. Therefore, logarithms and exponentiation are mathematical inverses:

log_x(x^y) = y

just as multiplication and division are inverses:

12 * 4 / 4 = 12

Some Notation:

log(x) [common log; base 10]

ln(x) [natural log; base e]

lg(x) [binary log; base 2]

floor(lg(x)) [floor binary log; base 2; greatest power of 2 less than the binary log of x]

ceil(lg(x)) [ceiling binary log; base 2; lowest power of 2 greater than the binary log of x]

etc.

Examples:

lg(8) = 3 [I cannot divide 8 by 2 more than 3 times].

2³ = 8