UNIT 2. Number Sets

Unit 2. Number Sets - vocabulary.

2.1 Classification of Numbers.


































Complex Numbers

z = a + bi


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



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.


















Whole Numbers

and  their opposites


NB. 0 does not have the opposite.




















Whole Numbers

Natural Numbers plus 0


NB. Natural numbers cannot be negative.

Natural Numbers

(Counting Numbers)


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


   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.


A ratio is a relationship between two or more numbers.


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: 



                               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... 

Unit 2. Number Sets - reading.

Mathematical notation.

 Ishango bone.     Single and split tallies.

Definitions, pictures and articles for reading adapted from:

   http://en.wikipedia.org/wiki/Tally_stick  http://en.wikipedia.org/wiki/Ishango_bone

Unit 2. Number Sets - problems.

Problem #2.1  Classification of Numbers.  Write down one example of:

1)    an irrational number

2)    a natural number

3)    a rational number

4)    a positive number

5)    an integer

6)    a decimal

7)    a complex number

8)    an odd number

9)    a fraction

10) a whole number

11) an even number

12) an imaginary number

13) a real number

14) a negative number

Problem #2.2  Fractions.  

a)    Name the fractions:

5/7, 3/5, 3/1, 1/2, 4/8, 3/2, 2/4, 7/3, 1/6, 6/9, 5/20, 6/35, 42/100

2a + b – 8c / 45 – a; 6a + 93 / 75c

b)    Give one example of:

a vulgar fraction

a proper fraction


a complex fraction

a mixed number

an invisible denominator

an improper fraction

a number whose ratio is 3:2

c)     Find the LCD of 4/5 and 7/9.

Problem #2.3  Decimals. Decimal Fractions.  

a)    Write down a decimal fraction whose denominator is 100.

b)    Write down the decimal fraction whose the integer part is 73 and the fractional part is 294.

c)    Truncate 4.902 to two decimal places.

d)    Add trailing zero to 5.

e)    Convert 25/75 to a decimal.

f)     Convert 0.36 to a fraction.

g)    Write down one example of: a terminating decimal, a recurring decimal, an irrational decimal.