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.

Also, a. 1 = 1. a = 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.

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

= i2 = – is correct!

But   . = = is wrong!

Keeping this and the fact that i2 = -1 in mind, we multiply in much the same way that we do with real numbers.

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  i2 = -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.