Occurrence of 2 in Mathematics and other Fields

I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind.

Lord Kelvin

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How many hands do you have? Or, how many straight angles make one complete angle? Or, how many outcomes are there if we toss an unbiased coin? Or, how many states are there in an electric switch? If someone ask us these questions, certainly the answer is 2. Yes, you got it correctly! Here, we will discuss about the power of 2.

2 is the natural number following 1 and preceding 3. An integer is called even, if it is divisible by 2. It is the smallest prime, which is even. Number 2 has the unique property 2 + 2 = 2 × 2 = 22. The square root of 2 is an irrational number.

1.9999… converges to 2 and any fraction with denominator 2 always terminates. 2 is the only root of the equation x – 2 = 0. In complex numbers, 2 can be written as the product of (1 + i) and (1 – i), where i2 = –1.

2 distinct points determine a line in plane, 2 intersecting planes in space determine a line, any quadrilateral has 2 diagonals and many figures have 2 lines of symmetry. Euler’s theorem tells us that, in any polyhedron VE + F = 2, where V is the number of vertices, E the number of edges and F the number of faces. Exactly 2 tangents can be drawn from an exterior point to a given circle. There are exactly 2 foci of an Ellipse and Hyperbola. Every hyperbola has 2 asymptotes.

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The number 2 plays the central role to measure computer memory. In fact, any measurement of computer memory like GB, MB etc. are the power of 2 (for example, 1 GB = 1024 MB = 210 MB; 1 MB = 1024 KB = 210 KB etc).

Also, the following infinite geometric series converges to 2, as the first term is 1 and common ratio is 1/2:

2 is the first Sophie Germain prime (a prime number q is a Sophie Germain prime if 2q + 1 is also a prime), first Lucas number (the Lucas numbers are defined by L0 = 2; L1 = 1 and Ln = Ln − 1 + Ln − 2 if n > 1), first Ramanujan prime and the third Fibonacci number (the Fibonacci numbers are defined by F1 = 1; F2 = 1 and Fn = Fn − 1 + Fn − 2 if n > 2).

2 is the first Factorial prime (a prime number that is one less or one more than a factorial) and powers of 2 are central to the concept of Mersenne prime (is a prime that is one less than a power of 2. Actually, it is a prime of the form 2n − 1) and Fermat numbers (are positive integers of the form 2P + 1, where P = 2n and n is a non-negative integer).

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The geometric progression of the form 2n (more explicitly, 1, 2, 4, 8, 16, 32, . . .)  is important in number theory because the sum of the first n terms of this progression is a prime number. Also, this sum times the nth term is a perfect number.

2 symbolizes many of the basic dualities like true/ false, yes/ no, left/ right. Either “you love to play with numbers” or “you do not love to play with numbers”. In a normal electric switch, there are 2 states, ON and OFF. If we throw an unbiased coin, there are 2 outcomes, HEAD or TAIL.

2 is the number of solutions of the equation cos x = |x| and the area bounded by the curve y = sin x in the range 0 to π (see the following figures).

In set theory, any element x has exactly 2 possibilities – either it belongs to the set A or it does not. Any set with one element will have exactly 2 subsets. There is a unique Group of order 2, up to isomorphism, namely (Z2, +). Any group of 2 elements is Abelian. The smallest field has 2 elements. If X be a one element set, then (P(X), + , . , ) is a 2 element Boolean algebra.

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