2.1 Classification of Numbers.

P

O

S

I

T

I

V

E

+

vs.

-

N

E

G

A

T

I

V

E

Complex Numbers

z = a + bi

where

i² = -1

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit satisfying x² = -1. In this expression, a is the real part and b is the imaginary part of the complex number.

Real Numbers 

A real number is a value that represents a quantity along a continuous line. Real numbers can be thought of as points on an infinitely long line called the number line, where the points corresponding to integers are equally spaced. 

or: A real number is any number whose square is non-negative. 

X²  ≥ 0

Imaginary

Numbers 

An imaginary number is a number that can be written as a real number multiplied by the imaginary unit i , which is defined by its property         i² = -1.

Rational Numbers

A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero.

Irrational Numbers

An irrational number is any real number that cannot be expressed as a ratio p/q, where p and q are integers, and q is non-zero.

Π,  

e

odd

vs.

even

……

except

0

Integers

Whole Numbers

and  their opposites

-3,-2,-1,0,1,2,3…

NB. 0 does not have the opposite.

Fractions

fractions

decimals

0,5

Whole Numbers

Natural Numbers plus 0

0,1,2,3,…..

NB. Natural numbers cannot be negative.

Natural Numbers

(Counting Numbers)

1,2,3,….

NB. Natural Numbers cannot be negative

2.2  Fractions.

A fraction represents a part of a whole or, more generally,

any number of equal parts.

A (vulgar,common, simple) fraction represents division of the numerator by the denominator 

three over five, the numerator over the denominator

     two a plus three b minus four c plus seven all over six a b

 -       fraction bar, vinculum                   /    forward slash, solidus

 three wholes              one half                three halves              three fourths / three quarters

vulgar fraction

 A vulgar fraction (also known as a common fraction or simple fraction) is a rational number written as a/b or , where a and b are both integers.

complex fraction

 In a complex fraction, either the numerator, or the denominator, or both, is a fraction or a mixed number. To reduce a complex fraction to a simple fraction, treat the longest fraction line as representing division.  

proper fraction

improper fraction

mixed number

4 

                                                                                 reciprocals

   is the reciprocal of 

The product of a fraction and its reciprocal is 1.

The reciprocal is multiplicative inverse of a fraction.

Every fraction or integer, except for 0, has a reciprocal.

                                                                     “ invisible denominator “

Any integer can be written as a fraction with the number 1 as denominator.

1 is sometimes referred to as the invisible denominator.

                                                                                          ratios

A ratio is a relationship between two or more numbers.

2:1

The ratio of ... to ...

is 2 to 1.

The word “ratio” gives its name to Rational Numbers.

Lowest (Least) Common Denominator

The smallest positive integer that is a multiple of the denominators of the fractions.

2.3  Decimals

Decimal fractions

A decimal fraction is a fraction whose denominator is a power of ten.

  629/1000  =  0.629 

zero point six two nine

the point is called: a decimal point 

Its numerical value truncated to two decimal places is: 

0.63 

0.250

                               the integer part                                                             the fractional part -             

                             ( the integral part) -                    decimal                      the part from the decimal separat

                               the part to the left                      separator                        to the right

                              of the decimal separator.

                              In this case                                                                      In this case

                             leading zero is added.                                                       trailing zero is added.

Converting decimals into fractions and fractions into decimals.

0.25 = 1/4   vs.   1/4 = 0.25

Types of decimals. 

a terminating decimal -  It is a decimal that stops: 0.125

a repeating (recurring) decimal - It goes on forever repeating the same digits over and over:  0.2727272727... (recurring) = 0.27 or 0.3

NB:     0.2727272727... = 27/99  (27 is called: the prime factor)

an irrational decimal - It does not terminate or repeat:  π = 3. 14159... ; the square root of 2= 1.4142...