The Properties of Real Numbers

Now that we have an alphabet, let's establish the rules of grammar for the language of mathematics. All of these rules work for any real numbers, so we'll use our new variables to express numbers in general. You can use the rules for substituting for variables with any of these rules, just be sure to use different numbers for different variables.

There are several kinds of rules. There are axioms which are so basic of a rule they are simply taken for granted. There are laws that are the consequence of axioms and are true everywhere all the time. And there are theorems which are rules that have been accepted as true because they have been built from axioms and laws and other theorems. Some of them we've already seen.



































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Closure Property
Adding any two real numbers results in another real number.
Multiplying any two real numbers results in another real number.
Transiti! ve Property
If two numbers are equal to the same number, then they are equal to each other.	If a=c and b=c then a=b
Equality Axioms
If two numbers are equal and you add the same number to each the resulting numbers are equal.	If a=b then a+c=b+c.
If two numbers are equal and you mutliply each by the same number the resulting numbers are equal.	If a=b then ac=bc.
Commutative Property
When adding real numbers, the order doesn't matter.	a+b=b+a
When multiplying real numbers, the order doesn't matter.	ab=ba
Associative Property
When adding real numbers, the grouping doesn't matter.	a+(b+c)=(a+b)+c
When multiplying real numbers, the grouping doesn't matter.	a(bc)=(ab)c
Distri! butive Property
Multiplying a sum of re! al numbe rs by a real number is the same as multiplying each real number of the sum first then adding.	a(b+c)=ab+ac
This is a powerful property. It allows us to pull a common factor out of the parts of an equation and deal with it separately.
Identities
Adding zero to a real number doesn't change the number.	a+0=a
Multiplying a real number by 1 doesn't change the number.	a×1=a
Inverses
Adding the additive inverse of a real number to itself results in the identity.	a+(-a)=0=(-a)+a
Multiplying the reciprocal of a real number to itself results in the identity.	a(1/a)=1 if a≠0
Zero Principle
Multplying by zero results in zero.	a×0=0
If the product of two real numbers is zero, then one of them must h! ave been zero.	If ab=0 then a=0 or b=0
Trichotomy Property
If two real numbers are compared, the second number can have only one of three outcomes, either greater than, less than or equal to.	a<b, a>b or a=b
Definition a^-1	a^-1=1/a
Properties of Negatives
Multiplying a real number by -1 changes the number's sign.	(-1)a=-a
	-(-a)=a
For any negative product of two real numbers, the negative sign can be assigned to either one the numbers.	(-a)b=-ab=a(-b)
The product of two negative real numbers is equal to the product of the additive inverses of the two numbers.	(-a)(-b)=ab
Negating the difference of two real numbers changes to signs of each number.	-(a-b)=b-a
Properties of Quotients
For any two rational numbers, the first rat! ional nu mber is equal to the second rational number if the product of the numerator of the first and the denominator of the second is equal to the product of the numerator of the second and the denominator of the first	a/b=c/d if and only if ad=bc
Mutliplying a rational number by a rational number equal to 1 changes the appearance of the rational number but not its value.	a/b=ad/bd
a/b+c/b=(a+c)/b
a/b+c/d=(ad+bc)/bd
a/b × c/d = ac/bd
a/b ÷ c/d = a/b × d/c = ad/bc
Definition
The principle square root of a number √a is the nonnegative real number b such that b squared is equal to a.	b²=a
Laws of Exponents
Definition
The absolute value of a real number is the magnitude of the number without a s! ign.	|a|=a, |-a|=a
Absolute Value Properties
The absolute value of a real number has the following properties
Positive Definite
For all real numbers, the absolute value of the number a is greater than or equal to 0 with |a|=0 only if a=0
Symmetric
For any two real numbers, |a-b|=|b-a|
Triangle Inequalities
For any two real numbers, the absolute value of the sum of the numbers is less than or equal to the sum of the absolute values of the numbers.	|a+b|≤|a|+|b|
For any two real numbers, the absolute value of the difference of the numbers is greater than or equal to the difference of the absolute values of the numbers.	|a-b|≥|a|-|b|
For any two real numbers, the absolute value of the difference of the absolute va! lues of the numbers is less than or equal to the absolute valu! e of the difference of the numbers.	||a|-|b||≤|a-b|

properties of real numbers