Archive for May, 2014

Exponential Expression

 

Exponential Expression

If a is a real number and n is a natural number,

 an  =  a .a .a ……a,

n factors of a

where n is the exponent, a is the base, and an is an exponential expression.

Exponents are also called powers.

 

For example, the product  2 . 2 . 2 . 2 . 2  is written

2 . 2 . 2 . 2 . 2 = 25.

5 factors of 2

The number 5 shows that 2 is used as a factor 5 times. The number 5 is the

exponent, and 2 is the base.

25   Exponent

 

Divide real numbers

Divide real numbers

 

The result of dividing one number by another is called the quotient. For example, when 45 is divided by 3, the quotient is 15. To define division of real numbers, we first write the quotient

of 45 and 3 as , which equals 15. The same answer will be obtained if

45 and are multiplied, as follows.

45 \div 3 = \frac{45}{3} = 45.\frac{1}{3}=15

This suggests the following definition of division of real numbers.

 

Division

For all real numbers a and b (where  b ≠ 0),

a\div b=\frac{a}{b}=a.\frac{1}{b}

In words, multiply the first number by the reciprocal of the second

 

Like signs :-The quotient of two nonzero real numbers with the same sign is positive.

Unlike signs :-The quotient of two nonzero real numbers with different signs is negative.

 

Find the reciprocal of a number

Find the reciprocal of a number

 

 Earlier, subtraction was defined in terms of addition. Now, division is defined in terms of multiplication. The definition of division depends on the idea of a multiplicative

inverse or reciprocal; two numbers are reciprocals if they have a product of 1.

Reciprocal

The reciprocal of a nonzero number a is  \frac{1}{a}.

 

Multiply real numbers

Multiply real numbers

 The answer to a multiplication problem is called the product. For example, 24 is the product of 8 and 3. The rules for finding signs of products of real numbers are given below.

 

Multiplying Real Numbers

 

Like signs:- The product of two numbers with the same sign is positive.

 

Unlike signs:-  The product of two numbers with different signs is negative.

Subtract real numbers

Subtract real numbers

 

 Recall that the answer to a subtraction problem is called the difference. Thus, the difference between 6 and 4 is 2. To see how subtraction should be defined, compare the following

two statements.

6 – 4 = 2

6 + (-4) = 2

 

Subtraction

 

For all real numbers a and b,

    a – b = a + (-b).

In words, change the sign of the second number (subtrahend) and add.

 

Adding Real Numbers

Adding Real Numbers

Like signs :- To add two numbers with the same sign, add their absolute

values. The sign of the answer (either + or -) is the same as the sign of

the two numbers.

 

Unlike signs  :- To add two numbers with different signs, subtract the

smaller absolute value from the larger. The sign of the answer is the

same as the sign of the number with the larger absolute value.

 

Use inequality symbols

Use inequality symbols

 

The statement 4 + 2 = 6 is

an equation; it states that two quantities are equal. The statement 4 ≠ 6

(read “4 is not equal to 6”) is an inequality, a statement that two quantities

are not equal. When two numbers are not equal, one must be less than the

other. The symbol <  means “is less than.” For example,

8 < 9, -6 <, 15, -6 < – 1, and 0 < \frac{4}{3} .

The symbol >  means “is greater than.” For example,

12 > 5, 9 > -2, and \frac{6}{5} > 0 .

In each case, the symbol “points” toward the smaller number.

 

Use absolute value.

 

Use absolute value.

Geometrically, the absolute value of a number a, written 0 a 0, is the distance on the number line from 0 to a. For example, the absolute value of 5 is the same as the absolute value of

– 5 because each number lies five units from 0.  That is,

\left | 5 \right | = 5 and \left | -5 \right | = 5.

Absolute value

 

CAUTION

Because absolute value represents distance, and distance is always positive

(or 0), the absolute value of a number is always positive (or 0).

 

The formal definition of absolute value follows.

Absolute Value

\left | a \right | \left\{\begin{matrix}  a& if& a& is& positive& or& 0    & \\  -a& if& a& is& negative    &  \end{matrix}\right.

 

The second part of this definition, \left | a \right | = –a if  a is negative, requires careful

thought. If a is a negative number, then  – a, the additive inverse or opposite

of a, is a positive number, so \left | a \right |  is positive.

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.

 

Sets of Numbers

Sets of Numbers

 

Natural numbers or counting numbers

 

 {1, 2, 3, 4, 5, 6, . . . }

 

Whole numbers  {0, 1, 2, 3, 4, 5, 6, . . . }

 

Integers  {. . . , 23, 22, 21, 0, 1, 2, 3, . . . }

 

Rational numbers  \left \{ \frac{p}{q} \mid p  and  q   are    integers, q   \neq  0   \right \}

Examples: , 1.3, or \frac{4}{3}.

 

Irrational numbers  {x | x is a real number that is not rational}

Examples: \sqrt{3}

 

Real numbers {x | x is represented by a point on a

number line}*