Basic properties of numbers

BASIC PROPERTIES LIST:

There are two fundamental operations, addition and multiplication, that have the following basic properties, listed with their commonly designated names (a,b,c  are real numbers):

(P1) Associative law for addition

a + (b + c) = (a+ b)+ c = a + b + c

(P2) Existence of additive identity

0 + a = a + 0 = a

(P3) Existence of additive inverse / negative

a + (-a) = (-a) + a = 0

(P4) Commutative law for addition

a + b = b + a

(P5) Associative law for multiplication

a(bc) = (ab)c = abc

(P6) Existence of multiplicative identity

1.a = a .1 = a

(P7) Existence of multiplicative inverse / reciprocal

a.a-1 = a-1.a = 1

(P8) Commutative law for multiplication

ab = ba

(P9) Distributive law

a(b + c) = ab + ac

(P10) Trichotomy law

For every number a, one and only one of the following holds:
(1) a is 0
(2) a is positive (a > 0 or a belongs to R+)
(3) -a is positive (-a belongs to R+)

(P11) Closure under addition

If a and b are positive ( a, b belong to R+), a+b is positive ( belongs to R+)

(P12) Closure under multiplication

If a and b are positive (belong to R+), a.b is positive ( belongs to R+)

 

MORE DETAILED VIEW OF THE BASIC AND SOME DERIVED PROPERTIES, DEFINITIONS AND EXAMPLES:

Closure law: The sum a + b and the product a.b or ab are unique real numbers; more specifically if a and b are positive (belong to R+), then a + b and a . b are positive  
Commutative law: a + b = b + a: order does not matter in addition, ab = ba: order does not matter in multiplication.
Associative law: a + (b + c) = (a+ b)+ c = a + b + c: grouping does not matter in addition. a(bc) = (ab)c = abc: grouping does not matter in multiplication.
Distributive law: a(b+ c) = ab + ac; (a+ b)c = ac + bc: multiplication is distributive over addition.
Identity law: There is a number 0 with the property that 0 + a = a + 0 = a; There is a number 1 with the property that  1.a = a .1 = a

Inverse law: For any real number a there is a real number -a such that a + (-a) = (-a) + a = 0 For any nonzero real number a there is a real number a-1 such that aa-1 = a-1a = 1; -a is called the negative of a ; a-1 is called the reciprocal of a

example: Associative and commutative Laws: Simplify (2 + x) + 7.

(2 + x) + 7 = (x + 2) + 7              Commutative Law

= x + (2 + 7)                                  Associative Law

= x+9

example Show that (a+ b)(c+ d) = ac + ad + bc + bd.

(a+ b)(c + d) = a(c + d)+ b(c + d)           Distributive law

= ac + ad + bc + bd                                    Distributive Law

This rule is referred to as FOIL (First Outer Inner Last). 

Zero factor law: For every real number a, a . 0 = 0 ;  If a.b = 0, then either a = 0 or b = 0.

Law for negatives: -(-a) = a ; (-a)(-b) = ab ; -ab = (-a)b = a(-b) = -(-a)(-b) ;  (-1)a = -a ;

Definition of substraction: a -b = a + (-b)
Definition of division: a / b = a . b-l. therefore b-l = 1 . b-l = 1 / b ; Note that since the number 0 has no reciprocal, a / 0 is not defined.
Properties of division: -(a/b) = (-a)/b=a/(-b)= - (-a)/(-b) ;  (-a)/(-b) = a/b ; a/b=c/d  if and only if a.d = b.c ; a/b = (p.a)/(p.b) for any non-zero p

Ordering properties: The positive real numbers R+ have the following properties:

1. If a and b are in R+, then a + b and ab Are in R+

2. For every real number a, either a is in R+, or a is zero, or -a is in R+.

If a is in R+, a is called positive; if -a is in R+, a is called negative.

The number a is less than b, written a < b, if b -a is positive. Then b is greater than a,

written b > a. If a is either less than or equal to b, this is written a ≤ b. Then b is greater

than or equal to a, written b ≥ a.

example: 2 < 3 because 3 - 2 = 1 is positive. -6 < 4 because 4 -(-6) = 10 is positive.

The following may be deduced from these definitions:

a > 0 if and only if a is positive.

If a ≠ 0, then a2> 0.

If a<b, then a+c < b+c.

If a < b, then  a.c > b.c if c < 0 and a.c < bc if c > 0

For any real number a, either a > 0, or a = 0, or a < 0.

If a < b and b < c, then a < c.

 

The real number line: Real numbers may be represented by points on a line in such a way that to each real number a there corresponds exactly one point on I, and vice versa:


 

Absolute value of a number: The absolute value of a, written lal, is defined as:

|a| = (  a if a ≥ 0; -a if a < 0)

example: |10| = 10 ; |-37.98| = 37.98 


Properties of absolute value:

|a| ≥ 0

|ab|=|a| |b|

|-a|=|a|

|a/b|=|a|/|b|

|a+b||a|+|b|


Complex numbers: The set C of numbers of the form a + bi, where a and b are real and i2 = -1, is called the complex numbers. Since every real number x can be written as x + 0.i, it follows that every real number is also a complex number. 

example: 3 + 2i, -5i  are examples of nonreal complex numbers.

 

Fractions properties: The fraction properties follow from the basic properties and the definition of division. The most commonly used fraction properties are listed below:
 
Order of operations: In expressions involving multiple operations of different types, the fol-

lowing order is followed:

1. operations within grouping symbols are performed first. If grouping symbols are nested

inside other grouping symbols, operations are performed in order from the innermost outward.

2. exponents are performed before multiplications and divisions (unless rule 1 indicate otherwise)

3. perform multiplications and divisions from left to right, before

additions and subtractions, also from left to right) ( unless unless rule 1 and 2 indicate

otherwise).

The above rules for the order of operations are known under the abbreviation PEMDAS - Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction

 

example:   Evaluate 10 - 8[(23 – 6)(3 + 4)]

10 - 8[(23 – 6)(3 + 4)]  = 10 - 8[(17) . (7) ] = 10 – 8.[ 119]
= 10 - 952 = -942
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