Number System, Fractions

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Basic numeracy

Number system

The Natural Numbers

The natural (or counting) numbers are 1,2,3,4,5, etc. There are infinitely many natural numbers. The set of natural numbers, {1,2,3,4,5,…}, is sometimes written N for short.  

The whole numbers are the natural numbers together with 0.  

(Note: a few textbooks disagree and say the natural numbers include 0.)  

The sum of any two natural numbers is also a natural number (for example, 4+2000=2004), and the product of any two natural numbers is a natural number (4×2000=8000). This is not true for subtraction and division, though.

The integers are the set of real numbers consisting of the natural numbers, their additive inverses and zero.


The set of integers is sometimes written JJ or ZZ for short.

The sum, product, and difference of any two integers is also an integer. But this is not true for division… just try 1÷21÷2.


Rational Numbers

The Rational Numbers The rational numbers are those numbers which can be expressed as a ratio between two integers. For example, the fractions 1 / 3 and

−1111 / 8 are both rational numbers. All the integers are included in the rational numbers, since any integer z can be written as the ratio z /1.


All decimals which terminate are rational numbers (since 8.27 can be written as 827 /100.) Decimals which have a repeating pattern after some point are also rationals:

for example


0.0833333… = 1 /12

The set of rational numbers is closed under all four basic operations, that is, given any two rational numbers, their sum, difference, product, and quotient is also a rational number (as long as we don’t divide by 0).


Irrational Numbers

The Irrational Numbers An irrational number is a number that cannot be written as a ratio (or fraction).  In decimal form, it never ends or repeats. The ancient Greeks discovered that not all numbers are rational; there are equations that cannot be solved using ratios of integers.


The Real Numbers

 The real numbers is the set of numbers containing all of the rational numbers and all of the irrational numbers.  The real numbers are “all the numbers” on the number line.  There are infinitely many real numbers just as there are infinitely many numbers in each of the other sets of numbers.  But, it can be proved that the infinity of the real numbers is a bigger infinity.

The Complex Numbers

The complex numbers are the set {a+bia+bi | aa and bb are real numbers}, where ii is the imaginary unit, −1−−−√−1.





Numbers and their relations






 1/4                                 1/2                       3/8


The top number says how many slices we have.



Equivalent Fractions Some fractions may look different, but are really the same, for example:



        4/8  =                           2/4    =                        1/2                      


Numerator / Denominator

We call the top number the Numerator, it is the number of parts we have.
We call the bottom number the 
Denominator, it is the number of parts the whole is divided into.




Adding fractions




       1/4              +             1/4                        =          1/2




Subtracting Fractions

Step 1. Make sure the bottom numbers (the denominators) are the same

Step 2. Subtract the top numbers (the numerators). Put the answer over the same denominator.

Step 3. Simplify the fraction



Example      : ¾  −  ¼ = ?

The bottom numbers are already the same

Subtract the top numbers and put the answer over the same denominator.

3/4 – ¼  = 3-1 /4 = 2/4

Simplify the fraction.

2/4= ½




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