## Posts Tagged ‘Numbers’

## Fundamental Property of Rational Numbers

**Fundamental Property of Rational Numbers**

** **

If is a rational number and if *c *is any nonzero real number, then

= .

In words, the numerator and denominator of a rational number may

either be multiplied or divided by the same nonzero number without

changing the value of the rational number.

## 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 such that

a. = 1 and .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.

## 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.

## 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.

## 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}*

## Complex Numbers Multiplication

**Complex Numbers Multiplication**

** **

The complex numbers obey the commutative, associative, and distributive laws. But although the property = dose not hold for complex numbers in general, it does hold when a = -1 and b is a positive real number.

To multiply square roots of negative real numbers, we first express them in terms if i .

For example,

. = . . . = i . i

= i^{2} = – is correct!

But . = = is wrong!

Keeping this and the fact that i^{2} = -1 in mind, we multiply in much the same way that we do with real numbers.

## Complex Numbers Addition and Subtraction

**Complex Numbers Addition and Subtraction**

The complex numbers follow the commutative and associative laws of addition. Thus we can add and subtract them as we do binomials with real-number coefficients , that is, we collect like terms.

Example

**a.** (8 + 6i) + (3 + 2i) = (8 + 3) + (6 + 2)I = 11 + 8i

**b.** (3 + 2i) – (5 – 2i) = (3 – 5) + [2 – (-2)]i = -2 + 4i

## The Complex Numbers Imaginary and Complex Numbers

**The Complex Numbers**

** Imaginary and Complex Numbers**

Negative numbers do not have square roots in the real-number system. However,

mathematicians have described a larger number system that contains

the real-number system, such that negative numbers have square roots. That

system is called the **complex-number system.**We begin by defining a number

that is a square root of -1. We call this new number *i*.

**The Complex Number i**

We define the number i to be . That is, i = and i^{2} = -1.

To express roots of negative numbers in terms of i we can use the fact that in the complex number, = = when p is a positive real number.

**Imaginary Number**

An imaginary* number is a number that can be named

bi,

where b is some real number and b ≠ o.

To from the system of complex numbers, we take the imaginary numbers and the real numbers and possible sums of real and imaginary numbers. These are complex numbers:

7 – 4i, i.