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What Are Interesting Terms In Mathematics?

What Is Pi?

Pi is the ratio of the circumference of any circle to the diameter of that circle. Regardless of the circle's size, this ratio will always equal pi. In decimal form, the value of pi is approximately 3.14. But pi is an irrational number, meaning that its decimal form neither ends (like 1/2 = 0.5) nor becomes repetitive (like 1/3 = 0.333333...). (To only 18 decimal places, pi is 3.141592653589793238.) Hence, it is useful to have shorthand for this ratio of circumference to diameter. According to Petr Beckmann's A History of Pi, the Greek letter π was first used for this purpose by William Jones in 1706, probably as an abbreviation of periphery, and became standard mathematical notation roughly 30 years later. (1 million digit of pi-click here)

What Is Infinity?

There is a popular illustration called the Hilbert’s paradox of the grand hotel.

Suppose Hilbert’s hotel has an infinite number of rooms and an infinite number of guests are booked into the hotel. By common sense, it seems like the hotel is fully booked right?Wrong. Infinite sets just defy logic. Suppose there was another guest who wanted to book into the hotel, all the hotel staff have to do is just shift guest in room number 1 to the next, the guest in room number two to the third and so on… So by this logic ∞+1=∞. Similarily∞−1=∞. Just remove the guest from room number 1 and shift the remaining guests to the predecessor of their room numbers. You still have an infinite number of guests.

Mathematical equations with infinity are not impossible to do. However, one must remember that infinity is not able to be exhausted. Because of this, there are some weird mathematical rules.
1. Infinity cannot become less than a value of one infinity. There is no such thing as half infinity.
2. Infinity added to or subtracted by itself is infinity. It cannot be 0.
3. Infinity added/subtracted by any number (let's say infinity-100) is simultaneously the equation, and infinity.
4. Infinity CAN be divided or multiplied. But each division or multiplication either results in a split or merge of infinity. That is, if I were to say all negative numbers, this would be -infinity. If I were to divide by all even positive numbers (infinity/2), each group would be infinite.
5. Infinity can be divided or merged, but can never really become more than itself. Or less.

So to put this in perspective.

Infinity + infinity = infinity
Infinity - infinity = infinity
Infinity * infinity = infinity
Infinity/infinity= infinity
Infinity - 100 = (infinity-100) OR infinity
Infinity/2 = two infinite groups OR infinity
Infinity*4= infinity (see rule 1)
(Seven infinite groups) * 7 = infinity
(Two infinite groups) * 10000 = infinity (see rule 5)

Viewed in this way, infinity is a sort of constant that you could plop variables into, but the tendency to consume everything makes in infeasible to use.

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What Is Imaginary Number?

The imaginary number is the quantity of the form ix, where x is a real number and i is the positive square root of -1. The term "imaginary" probably originated from the fact that there is no real number z that satisfies the equationz2 = -1. But imaginary numbers are no less "real" than real numbers. The quantity i is called the unit imaginary number.

The unit imaginary number has some intriguing properties.  For example:

(-i)2 = -1
but -i is different from i

i3 = i2i = (-1)i = -i

i4 = i2i2 = (-1)(-1) = 1

i5 = i3i2 = (i3)(-1) = (-i)(-1) = i

in = i(n-4)

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