Binomial Number. Numbers. Real Number Sets Real Number Line Real Number Venn Diagram Complex Number Sets Complex Number Plane z = x + iy, i = √−1 Complex Number Venn Diagram Properties of the Number Sets Closed under addition (multiplication, subtraction, division) means the sum (product, difference, quotient) of any two numbers in the set is also in the set.Dense: Between any two numbers there is another number in the set.Continuous with no gaps. The complex numbers are the algebraic completion of the real numbers. Infinity, ∞ The integers, rational numbers, and algebraic numbers are countably infinite, meaning there is a one-to-one correspondence with the counting numbers. PDF format for printing PDF | Terms of Use | Buy Poster. Archive.
Frivolous Theorem of Arithmetic. Happy number. A happy number is a number defined by the following process: Starting with any positive integer, replace the number by the sum of the squares of its digits, and repeat the process until the number equals 1 (where it will stay), or it loops endlessly in a cycle which does not include 1. Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers (or sad numbers).[1] Overview[edit] More formally, given a number , define a sequence , ... where is the sum of the squares of the digits of . If a number is happy, then all members of its sequence are happy; if a number is unhappy, all members of the sequence are unhappy. For example, 19 is happy, as the associated sequence is: The 143 happy numbers up to 1,000 are: The happiness of a number is preserved by rearranging the digits, and by inserting or removing any number of zeros anywhere in the number.
Sequence behavior[edit] If n is not happy, then its sequence does not go to 1. , or For and above, Question Corner -- Why is e^(pi*i) = -1? Navigation Panel: (These buttons explained below) Question Corner and Discussion Area Asked by Brad Peterson, student, Roy High on January 29, 1997: I was watching an episode of The Simpsons the other day, the one where Homer gets sucked into the third dimension, and in this 3-D world, there was an equation that said . So I put it into the calculator and it worked, but I have no idea why, because e to any power isnt supposed to be a negative number, and I thought pi was in no way related to e.
We'd be glad to explain; that's exactly what this area is here for. The first question to ask, though, is not "why does ", but rather, "what does even mean? " Once that question is answered, it will be much more clear why . For all x, a fact which is known as de Moivre's formula, and illustrates how closely related the exponential function is to the trigonometric functions. So now, the question is, why is the "right" thing to define what e raised to an imaginary power means? (i.e., a); in other words, . If . No, really, pi is wrong: The Tau Manifesto by Michael Hartl | Tau Day, 2010.