## Archive for May, 2014

## Exponential Expression

**Exponential Expression**

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

** a^{n} **=

*a .*

*a .*

*a ……*

*a***,**

*n *factors of *a*

where *n *is the **exponent, ***a *is the **base, **and *a ^{n} *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 **= **2 ^{5}**.

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.**

**2 ^{5} ←**

**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.

This suggests the following definition of division of real numbers.

**Division**

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

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 .

## 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 < .

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

12 > 5, 9 > -2, and .

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,

**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

The second part of this definition, = –*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 is positive.

## Additive Inverse

**Additive Inverse**

For any real number *a*, the number 2*a *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 p and q are integers,**

*Examples: *, 1.3, or .

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

*Examples: *

* *

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

number line}*