swiss-algebra-help.com # Prime Numbers

A special case of algebra factoring

is when we don't find many factors for a number but just two: 1 and the number itself:

For example:

2 = 1 x 2 = 1 + 1

3= 1 x 3 = 1 + 1 + 1

5= 1 x 5 = 1 + 1 + 1 + 1 + 1

7= 1 x 7 = 1 + 1 + 1 + 1 + 1 + 1 + 1

Well, these special numbers are called prime numbers.

All other numbers which do have more factors are called composite numbers instead.

A (positive) integer larger than 1 that has only two factors, itself and 1 is called prime number.

e.g.: 2, 3, 5, 7, 11, 13, 17, 23, 27.....(for more prime numbers click here )

By definition, the number 1 is NOT a prime number

Prime numbers are very important numbers because, (that's really fascinating!) all whole numbers of the universe are either prime numbers or are built by them!

So, they are like little gems, building up a numerical mathematical skeleton.

To find out which ones they are, we must proceed with a (simple) procedure (an algorithm) which keep asking us the ordered sequence of divisibility rules of an "in crescendo" (upwards) series of prime numbers.... 2,3,5,7,11,13,17,23, etc...

Yes divisibility by 2,3,5,7,11,13,17,23, etc...

This is called factorization.

It's easier to illustrate this concept with an example.

But beforehand, you should but refresh the divisibility rules. Moreover, you can also make use of this software which I tried out and found quite well made... despite the usual American (commercial) idea of forcing fun in learning... (and the voices are ugly, the interactivity can also be improved, but the content is ok). Provided you really use it without getting distracted....

### Prime factorization for the number 36 36 is divisible by 2 ---------------------> 36 : 2 = 18 18 is divisible by 2 ---------------------> 18 : 2 = 9 9 is divisible by 3 -----------------------> 9 : 3 = 3 3 is divisible by 3 only (prime) --------> 3 : 3 = 1

so,

We can see that we used the factor "2" two times (36:2 and 18:2) and the factor "3" 2 times (9:3 and 3:3). In fact 36 = 2x2x3x3. We can also write this (we'll come later to this strange way of mathematical writing convention, called "exponentials") as: What we did is the prime factorization of the number 36

Alternatively you can check my factorization calculator webpage (to be used as a way to control a result, NOT as a a shortcut from studying!).

1708. The laws of algebra, though suggested by arithmetic,
do not depend on it. They depend entirely on the conventions
by which it is stated that certain modes of grouping the symbols
are to be considered as identical. This assigns certain prop-
erties to the marks which form the symbols of algebra. The
laws regulating the manipulation of algebraic symbols are
identical with those of arithmetic. It follows that no algebraic
theorem can ever contradict any result which could be arrived at
by arithmetic; for the reasoning in both cases merely applies
the same general laws to different classes of things. If an
algebraic theorem can be interpreted in arithmetic, the cor-
responding arithmetical theorem is therefore true.

Universal Algebra (Cambridge, 1898), p. 2.

1709. That a formal science like algebra, the creation of our
abstract thought, should thus, in a sense, dictate the laws of its
own being, is very remarkable. It has required the experience
of centuries for us to realize the full force of this appeal.

MATHEWS, G. B.

F. Spencer: Chapters on Aims and Practice of
Teaching (London, 1899), p. 184.

ALGEBRA 277

1710. The rules of algebra may be investigated by its own
principles, without any aid from geometry; and although in
many cases the two sciences may serve to illustrate each other,
there is not now the least necessity in the more elementary
parts to call in the aid of the latter in expounding the former.

source: Memorabilia mathematica; or, The philomath's quotation-book - Moritz, Robert Édouard, 1868-1940

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