Prime Numbers – multiplication vs. addition

Prime Numbers – multiplication vs. addition

This post is meant to discuss why multiplication is used to define prime numbers and composites instead of addition.

Let us say we want to compose the number 6 by multiplication of other positive whole numbers. There are 2 ways to do this.

  1.  6×1=6
  2.  2×3=6

In terms of 6 there are smaller whole numbers that can be multiplied to compose or produce the number 6. Since multiplication obeys both the associative and commutative property, we won’t concern ourselves with the alternative order ( i. e., 1×6 and 3×2). Considering the number 6 there is only one way to make the number by multiplication by the primes (please note this post is not going into detail regarding ° what the primes are).

Consider the number 90. There are several ways to produce this by multiplication.

  1.  2×45
  2.  2×5×9
  3.  6×15

These are a few ways and there are others. However, there is only 1 way to make 90 by multiplication of primes.

  1. 2×3×3×5

Again, order is not important.

Primes are numbers that cannot be made by multiplication of other whole numbers. The first few primes are shown below.

First Few Primes

  • 2
  • 3
  • 5
  • 7
  • 9
  • 11
  • 13
  • 17
  • 19
  • 23
  • 29
  • 31
  • 37
  • ad infinitum

So one may ask,

why not define a prime-kind of number from addition?

The reason is shown above. When considering multiplication we only have 1 way to make A number with primes. However, addition provides a plethora of ways to produce any whole number using whole numbers. Here we will also exclude negative numbers for the sake of consistency.

Consider the number 6 again. If one want to make 6 from other whole numbers, we have many choices. A few examples are:

  1. 5+1
  2. 2+3+1
  3. 3+3
  4. 2+2+2
  5. 1+1+1+1+1+1

There are others. Every number along the way Can be produced by addition of other whole numbers except 1. Thus, there is no obvious way to compose other numbers that exploits some unique process (other than addition of 1 which gave rise to the sequence called the natural numbers).

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