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Lecture-1181.pptx

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Review of Fundamental Mathematics

Chu V. Nguyen

Algebraic Numbers

Each algebraic number (hence forth, referred to as a number) consists of two components. A POSITIVE numeric value which can be expressed as:

2, 5, 8; or in notation such as a, b, x, y, z; or

and an algebraic sign in front:

+ (plus) or – (minus)

Usually, the + (plus) sign is omitted; for examples:

+2 is often expressed as 2, or +z is written as z; and as

Basic Properties of Algebraic Numbers

Closure Properties:

The sum or product of two algebraic numbers is an algebraic number; thus,

+a-x or a-x or, -z-a is an algebraic number.

and

(+a)*(+b), or a*b, (a)(-x), or (-z)*(-a) is also an algebraic number.

Note: a.b may be expressed in a number of other ways, such as

(a)(b), (a)(b), a(b), (a)b or just ab.

“*” indicates “multiplication.”

Basic Properties of Algebraic Numbers (cont.)

Commutative Properties:

The sum or product of two numbers is not affected by the order in which they are combined. That is, for numbers a and z

a + z = z + a, or a +(-z) = -z + a

a*b = b*a , or (a)(-z) = (-z)(a).

Examples:

3 + 5 = 5 + 3 = 8 or 6 + (-4) = 6 – 4 = 2.

3*4 = 4*3 =12 or 5(-4) = (-4)*5 = -20

Basic Properties of Algebraic Numbers (cont.)

Associative Properties:

The sum or product of three numbers is the same when the third is combined with the first two or when the first is combined with the last two . That is, for numbers a, y and z

(a + y) + z = a + ( y + z), or (a + y) - z = a + (y -z)

(a*y)z = a( yz), or (ay)(- z) = a[y(-z)]

Basic Properties of Algebraic Numbers (cont.)

Distributive Properties:

The product of a number times the sum or the difference of two others is the same as the sum or the difference of the products of the first number times each of the others. That is, for numbers a, y and z

a(y+ z) = ay + az, or a(y – z) = ay + a(-z) = ay –az.

This may also be written in the form:

(y + z)a = ya + za, or (y – z)a = y + (-z)a = ya –za.

Basic Properties of Algebraic Numbers (cont.)

Identity Properties:

The sum of any number x and a zero is the given number, x.

0 + x = x + 0 = x

We call zero the additive identity.

The product of any number x and 1 is the given number, x.

1*x = x*1 = x

We define 1 as the multiplicative identity.

Basic Properties of Algebraic Numbers (cont.)

Inverse Properties:

For each number x, there exists another number, -x, such that the sum of x and –x is zero.

x + (-x) = 0

We define –x or (-x) as the additive inverse or opposite of x.

For each number x, different from zero, there exists another number ,

such that the product of x and is 1.

x* = *x = 1

We call the multiplicative inverse or reciprocal of x.

Basic Properties of Algebraic Numbers (cont.)

Other Inverse Properties:

The opposite of the opposite of a number z is the number z

-(-z) = z

The opposite of the sum is the sum of the opposites. That is , for numbers q and z

-(q + z) = (-q) + (-z)

The opposite of a product of two numbers is the product of one number times the opposite of the other. That is for two numbers a and b

-(ab) = (-a)b = a(-b)

Basic Properties of Algebraic Numbers (cont.)

Properties of zero:

The product of a number p and zero is zero.

p*0 = 0*p = 0.

It is also important to note that for any two numbers x and y

If x*y = 0, then either x = 0, or y = 0 or both x = 0 and y = 0.

Subtraction and Addition

The difference 0r sum of two numbers x and z is defined as

x – z = x + (-z).

For examples,

5 - 8 = 5 + (-8) = -3.

or 5 – (-8) = 5 + 8 = 13

Alternatively, we can say that for three numbers a, b, and c

a – b = c if and only if a - c = b

Thus 5-2 = 3 because 5 - 3 = 2

Note: “if and only if” is sometime written as “iff.”

Multiplication and Division

The product 0r the quotient of two numbers a and z is defined as

(a)(– z) or a(-z).

; z must be non-zero, since we cannot divide by zero

Signs:

The product and the quotient of two numbers, a and z will have a positive ( + or plus) sign if a and z have the same sign.

The product and the quotient of two numbers a and z will have a negative ( - or minus) sign if a and z have opposite signs.

Practices

Evaluate the following expressions:

(i) 2(-3)

(ii) -5(x)

(iii) (5*6)(7)

2. Find the additive inverse for 4.

3. Find the multiplicative inverse for 2

4. Find the values for the following expressions:

(i) 8 – (-4)

(ii) 2 + (-5)