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Free Homework Math Help: Properties of Real Numbers

The Properties of the Real Numbers

Real numbers have properties that they conform to when they are used in math operations. Many of these rules form the foundation for doing homework problems. Understanding these properties can help with algebra and other math.

Table of Real Number Properties, Rules and Formulas

a, b and c are real numbers.

Property Name Expression
Reflexive a = a
Symmetric If a = b, then b = a
Transitive If a = b and b = c, then a = c
Closure, Addition The sum of a and b is a real number, a + b
Closure, Product The product of a and b is a real number a*b or ab
Commutative Addition a + b = b + a
Commutative Multiplication ab = ba
Distributive a(b+c) = ab+ac or (b+c)a = ba+ca
Identity, Addition a+0 = 0+a = a
Identity, Multiplication a*1 = 1*a = a
Inverse, Addition a+(-a) = -a+a = 0
Inverse, Multiplication
a*1
a		   =   1
a		*a   =   1,    a≠0
Substitution If a = b, then a can be substituted for b and vice versa
Addition If a=b, then a+c = b+c
Multiplication If a=b, then ac = bc
Subtraction a-b = a+(-b)
Division
1
b		*a   =   a*1
b		   =   ab-1
Division by Zero
a
0		    is undefined
Zero, Multiplication a*0 = 0
Zero, Division
0
a		   =   0,    a≠0
Negatives (-a)(-b) = ab
Fractions, Equality
a
b		   =   c
d		   if and only if ad = bc
Fractions, Equivalent
a
b		   =   ac
bc		,   c≠0
Fractions, Addition, Same Denominator
a
b		 + c
b		   =   a+c
b
Fractions, Addition, Different Denominator
a
b		 + c
d		   =   ad+bc
bd
Fractions, Subtraction, Same Denominator
a
b		 - c
b		   =   a-c
b
Fractions, Subtraction, Different Denominator
a
b		 - c
d		   =   ad-bc
bd
Fractions, Multiplication
a
b		 * c
d		   =   ac
bd
Fractions, Division
a
b		 ÷ c
d		   =   a
b		 * d
c		   =   ad
bc		,   b≠0, c≠0
Fractions, Negative
-a
b		   =   -a
b		   =   a
-b
Fractions, Double Negative
-a
-b		   =   a
b

By understanding these basic rules, it is possible to do basic algebra. Some of these properties are also included in earlier math. Some of these rules can also be extended such as the rule for distribution. For example, a(b+c+d+e+...+z) = ab+ac+ad+ae+...+az. In algebra, the concept of foil is based on the distributive property. When adding and subtracting fractions, the LCD or least common denominator can be used also. Sometimes, the multiplicative identity is used to make the denominators the same.