Circles – history and more

I became interested in exploring circles while helping out my niece with some problems involving arcs. So, I am just collecting here some interesting resources found online.

1) Why is a Circle 360 Degrees – Many of us have wrong notion that a circle is just 360 degrees . “A circle has 360 degrees, but it also has 400 gradients and approximately 6.2831853 radians. It all depends on what *units* you measure your angles with.”

2) How Egyptian’s measured circle’s area

3) In this blog, the author says Circles Are the Mother of All Inventions. Really? The author says ” A circle is not a “thing-in-itself.” It is a semantic fabrication that exists only in our imagination”

4) The Beauty of Circles and Spheres

Intuition behind why 2^-1 is 1\2

^ symbol means exponent

What is 2^3 ? It is 8.
What is 2^2 ? It is 4.
What is 2^1? It is 2.

So, we can observe, that, as the exponent decreases from 3 -> 2 -> 1, each result is obtained by dividing by 2 – 8, 4 (8\2), 2 (4\2).

So, what is 2^0? Divide 2\2. And, the answer is 1 !!.

And, so what is 2^-1 ? Divide 1\2.!!

I learned this at Khan Academy

Another way of looking at negative exponents is,

A negative exponent means how many times to divide by the number.

Beauty – Proof by Contradiction

What is Proof By Contradiction -

To prove something by contradiction, we assume that what we want to prove is not true, and then show that the consequences of this are not possible. That is, the consequences contradict either what we have just assumed, or something we already know to be true (or, indeed, both) – we call this a contradiction.

Axioms

An Axiom is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other (theory dependent) truths.

1. Additive Identity
[ a + 0 = a ]

2. Multiplicative Identity
[ a . 1 = 1 ]

3. [ 1 != 0 ]

4. Additive Inverse
[ a + (-a) = 0 ]
or, it could be written as a – a =0

5. Multiplicative Inverse
[ a. 1/a = 1 ]

6. [ a .0 = 0 ] (this axiom has a proof)

We all know that we cannot divide by 0. Can this truth(which we simply accept) be proved?

We will start by assuming we can divide 1 by 0.

From Axiom 5, the multiplicative inverse, for any n

n. 1/n = 1.
So, when n = 0 , then
0 . 1/0 = 1

Lets choose 2 numbers, x and y, such that x != y.
From axiom 6, we can say

x . 0 = 0
y . 0 = 0
x . 0 = y . 0

Lets, multiply each side by inverse of 0 i.e. 1/0.

(x.0) 1/0 = (y.0) 1/0
We can rewrite it,
x(0 . 1/0) = y (0 . 1/0)

Now, 0. 1/0 is 1(multiplicative inverse), which means
x = y , but this contradicts our assumption that x!=y.
So, this means it was wrong to assume that 1/0 exists.

There are some famous proofs using this method -
1. There are infinite prime numbers ( Euclid proved this in Elements)
2. √2 is irrational

Sources:
1. Proof by Contradiction
2. The Art Of The Infinite ( our lost language of numbers)