Posts Tagged ‘Inverse’

Inverse Variation

Inverse Variation

 

y  varies inversely as x if there exists a real number k such that

 

y = \frac{k}{\dot{a}}

 

Also, y varies inversely as the nth power of x if there exists a real

number k such that

   y = \frac{k}{\dot{x^{n}}}

Properties of Real Numbers

Properties of Real Numbers

 

Distributive Property

 

For any real numbers a, b, and c,

 

a(b + c) = ab + ac and (b + c)a = ba + ca.

 

 

Inverse Properties

 

For any real number a, there is a single real number – a such that

a + (-a) = 0 and –a + a = 0.

The inverse “undoes” addition with the result 0.

For any nonzero real number a , there is a single real number \frac{1}{a} such that

a.\frac{1}{a} = 1 and  \frac{1}{a}.a = 1.

The inverse “undoes” multiplication with the result 1.

 

 

Identity Properties

 

For any real number a , a + 0 = 0 + a = a.

Start with a number a; add 0. The answer is “identical” to a.

Also, a. 1 = 1. a = a.

Start with a number a; multiply by 1. The answer is “identical ” to a.

 

Commutative and Associative Properties

For any real numbers a, b, and c,

a + b = b + a

and  ab = ba.

Commutative properties

Interchange the order of the two terms or factors.

Also, a + (b + c) = (a + b) + c

and a(bc) = (ab)c.

Associative properties

Shift parentheses among the three terms or factors; order stays the same.

 

 

Additive Inverse

Additive Inverse

For any real number a, the number 2a is the additive inverse of a.

Change the sign of a number to get its additive inverse. The sum of a number

and its additive inverse is always 0.

 

The symbol “-” can be used to indicate any of the following:

a.    a negative number, such as -9 or -15;

 

b. the additive inverse of a number, as in “- 4 is the additive inverse of 4”;

 

c. subtraction, as in 12 – 3.

 

In the expression – (-5), the symbol “-” is being used in two ways: the

first – indicates the additive inverse of -5, and the second indicates a negative

number, -5. Since the additive inverse of – 5 is 5, then – (- 5) = 5. This

example suggests the following property.

 

 For any real number a,      – (a) = a.