Numerical Systems


Overview

In this article I’ll be covering some things that most individuals already know and some might not even be interested in.  There are a few reasons I’m doing this, but the most important one if for me to have a central place where I can refer to some arbitrary things that we know, but that we all do in fact forget and need a refresher on every now and again.  So, in this range of articles I’ll have some short summaries of various things including Mathematics, Networking, and Analysis & Design etc.  If you’re not interested, move onto one of those articles that do interest you. J  In this article I’ll move straight into the Number Systems where I’ll try and explain in a very brief way the difference of the different number systems with their respective Properties.  I’ll cover it in the following manner:

  1. Mind Map
  2. 11 Laws of Numbers
    • Law 1:  Commutativity
    • Law 2:  Associativity
    • Law 3:  Distributivity
    • Law 4:  Multiplicative Identity
    • Law 5:  Linearity
    • Law 6:  Monotonicity
    • Law 7:  Transitivity
    • Law 8:  Additive Identity
    • Law 9:  Absence of zero divisors
    • Law 10:  Additive Inverse
    • Law 11:  Multiplicative Inverse
  3. Number Systems
  4. Overview Table
  5. Summary

Mind Map

I find Mind Maps as an easy way to view and remember certain things.  I don’t use them as often as I should, but in this case I thought it wise to do so seeing that it gives us a very brief and high level overview of the numbers and the laws that pertain to them.

Number Systems - Mind Map

Number Systems - Mind Map

11 Laws of Numbers

Law 1:  Commutativity

Definition:           For all integers of m and n, m + n = n + m and m.n = n.m

Explanation:       You can switch the numbers on both sides of the operand (+ or .) and the answer will be the same.

Example:             2 + 4 = 4 + 2 = 6

2 x 4 = 4 x 2 = 8

Law 2:  Associativity

Definition:           For all integers of m, n and k, m + (n + k) = (m + n) + k and m(nk) = (mn)k

Explanation:       You can move the positions of the brackets and the answer will be the same.

Example:             2 + (4 + 3) = (2 + 4) + 3 = 9

2 x (4 x 3) = (2 x 4) x 3 = 24

Law 3:  Distributivity

Definition:           For all integers of m, n and k, m(n + k) = (mn) + (mk)

Explanation:       If you have a multiplier on the outside of the brackets (m) and you have an addition (+) happening on the inside of the bracket then you will get the same answer if you multiply the items (n and k) on the inside with the multiplier (m) and then do the addition.

Example:             2(4 + 3) = (2 x 4) + (2 x 3) = 14

Law 4:  Multiplicative Identity

Definition:           For all integers of m, m x 1 = m

Explanation:       If you multiply any number (m) with 1, the answer will be the original number (m).

Example:             4 x 1 = 4

Law 5:  Linearity

Definition:           For all integers of m and n, exactly one of the following statements is true

m < n

m = n

m > n

Explanation:       A number can only be smaller than, equal to or bigger than any other number.

Example:              2 < 4

2 = 2

2 > 1

Law 6:  Monotonicity

Definition:           For all integers of m, n and k

if m = n, then m + k = n + k and m.k = k.m and

if m < n then m + k < n + k and

if k > 0, then m.k < n.k while

if k < 0, then m.k > n.k

Explanation:       If you look at the logic above then you may think that this is the hardest law of them all.  It’s not, so look again and look at it in the smallest pieces.  If you look at the first section where m = n, you’ll see that this section is the same as Law 1 (Commutativity).  Now if you look at the second section where m < n then you’ll see that it’s the same as stating that m < n if you take k out of the picture while doing addition.  The third section states the same as the previous section, but with k not being able to be zero seeing that anything multiplied by 0 is in fact then 0.

Example:             if m = 2 and n = m then

m + k = n + k

2 + k = 2 + k (substitution)

if m = 2 and n = 3 then

m < n and therefore

m + k < n + k

2 + k < 3 + k (Substitution)

if m = 2 and n = 3 and k = 4 then

m < n and therefore

m.k < n.k

2 x 4 < 3 x 4 (Substitution)

8 < 12

Law 7:  Transitivity

Definition:           For all integers of m, n and k,

if m = n and n = k then m = k and

if m < n and n < k then m < k

Explanation:      If m is smaller than n and n is smaller than k then m is also smaller than k.

Example:             if m = 1 and n = 2 and k = 3 then 1 < 2 < 3

Law 8:  Additive Identity

Definition:           For all integers of m, m + 0 = m

Explanation:       If you add zero to any number the answer will be the original number.  In this case the Additive Identity is the number zero.

Example:             1 + 0 = 1

Law 9:  Absence of zero divisors

Definition:           For all integers of m and n, m.n = 0 if and only if m = 0 or n = 0

Explanation:       If you multiply zero to any number the answer will be the zero.

Example:             2 x 0 = 0

Law 10:  Additive Inverse

Definition:           For all positive integers m there exists an integer n such that, m + n = 0

Explanation:       Every number has an opposite number that exists where if the two were added to each other the answer would be zero.

Example:             if m = 2 and n = -2 then m + n = 2 + -2 = 0

Law 11:  Multiplicative Inverse

Definition:           For every non-zero rational number x there exists a rational number y such that, x.y = 1

Explanation:       This is the same kind of logic as with Law 10 (Additive Inverse) but over multiplication.  So, in fact it states that every number has a number where if the two were multiplied with each other the answer would be 1.

Example:             if x = 3/7 and y = 7/3 then x.y = (3/7) x (7/3) = 1

Number Systems

Positive Number:  Z+

Definition:  All positive integers i.e. numbers greater than zero i.e. n > 0.
Symbol:  Z+
Example:  {1, 2, 3, 4 …}

Applicable Laws:               Law 1:  Commutative

Law 2:  Associative

Law 3:  Distributive

Law 4:  Multiplicative Identity

Law 5:  Linearity

Law 6:  Monotonicity

Law 7:  Transitivity

Law 8:  Additive Identity

Natural Number:  N

Definition:  All positive integers i.e. numbers greater including zero i.e. n >= 0.
Symbol:  N
Example:  {0, 1, 2, 3, 4 …}

Applicable Laws:               Law 1:  Commutative

Law 2:  Associative

Law 3:  Distributive

Law 4:  Multiplicative Identity

Law 5:  Linearity

Law 6:  Monotonicity

Law 7:  Transitivity

Law 8:  Additive Identity

Law 9:  Absence of zero divisors

Integer:  Z

Definition:  All integers including the numbers less than zero i.e. n >= 0 and n <= 0.
Symbol:  Z
Example:  {…, -2, -1, 0, 1, 2 …}

Applicable Laws:               Law 1:  Commutative

Law 2:  Associative

Law 3:  Distributive

Law 4:  Multiplicative Identity

Law 5:  Linearity

Law 6:  Monotonicity

Law 7:  Transitivity

Law 8:  Additive Identity

Law 9:  Absence of zero divisors

Law 10:  Additive Inverses

Rational Number:  Q

Definition:  All numbers of the form p/q where p and q are integers and q is non-zero.
Symbol:  Q
Example:  1/2, 1/3, 1/4 …

Applicable Laws:               Law 1:  Commutative

Law 2:  Associative

Law 3:  Distributive

Law 4:  Multiplicative Identity

Law 5:  Linearity

Law 6:  Monotonicity

Law 7:  Transitivity

Law 8:  Additive Identity

Law 9:  Absence of zero divisors

Law 10:  Additive Inverses

Law 11:  Multiplicative Inverses

Irrational Number:  ‘Q

Note:  Irrational Numbers are not seen as a Number system seeing that it not closed as the number systems are.  These numbers are those numbers that have no end like pi or the answer of 22/7 symbolized by the sign – ∏.
Definition:  Any Real Number which cannot be expressed as a fraction a/b, where a and b are integers, with b being non-zero, and is therefore not a Rational Number.
Symbol:  ‘Q (I’m only adding a symbol to this number for personal notes.  In fact Irrational Numbers have no symbol.)
Example:  √2
Further Reading:  Wikipedia

Applicable Laws:               Law 1:  Commutative

Law 2:  Associative

Law 3:  Distributive

Law 4:  Multiplicative Identity

Law 5:  Linearity

Law 6:  Monotonicity

Law 7:  Transitivity

Law 8:  Additive Identity

Law 9:  Absence of zero divisors

Law 10:  Additive Inverses

Law 11:  Multiplicative Inverses

Real Number:  R

Definition:  The expanded number system that consists of the combination of the rational and irrational numbers.
Symbol:  R
Example:  Same as Q and ‘Q

Applicable Laws:               Law 1:  Commutative

Law 2:  Associative

Law 3:  Distributive

Law 4:  Multiplicative Identity

Law 5:  Linearity

Law 6:  Monotonicity

Law 7:  Transitivity

Law 8:  Additive Identity

Law 9:  Absence of zero divisors

Law 10:  Additive Inverses

Law 11:  Multiplicative Inverses

Overview Table

In this table you’ll see a short summary and yet another way to put what we already have above.  For some people (like me) it helps to have more than one way to see things.  I like short and concise, so here is a table showing the Number System and the properties that pertain to them.

Number Systems - Laws

Number Systems - Laws

Summary

This is the first of a series of articles which will have a very wide range.  I’m doing this seeing more for myself than anyone else, so if you’re not interested, please have a look at the other articles on this site.  I’m sure you’ll find something interesting and more applicable to you.  My next article in this range will be around Set Theory, which will include the basics of this field and which should also grow into something that I can use as reference point for the future.  As mentioned earlier in this article, I love math and love reading up about things.  Writing articles like this is one way for me to learn things and remember them and even though this was not much of a learning experience for me, I’m sure the articles that follow will be J  Everything has a base, and if you don’t have that base or if your understanding is a bit rough and outdated then you will get to a point where you’ll have to go back.  These kinds of articles are that base for me and will be used as reference point so as to always be a refresher in the most basic things.

I hope you enjoyed this one and if not, stay tuned as I start delving more and more into the interesting things 🙂

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