Number System

### Author Topic: Number System  (Read 2004 times)

#### Masuma Parvin

• Sr. Member
•    • Posts: 323 ##### Number System
« on: January 13, 2015, 04:22:34 PM »

THE REAL NUMBER SYSTEM

The real number system evolved over time by expanding the notion of what we mean by the word “number.” At first, “number” meant something you could count, like how many sheep a farmer owns. These are called the natural numbers, or sometimes the counting numbers.

Natural Numbers

or “Counting Numbers”
1, 2, 3, 4, 5, . . .

The use of three dots at the end of the list is a common mathematical notation to indicate that the list keeps going forever.
At some point, the idea of “zero” came to be considered as a number. If the farmer does not have any sheep, then the number of sheep that the farmer owns is zero. We call the set of natural numbers plus the number zero the whole numbers.

Whole Numbers

Natural Numbers together with “zero”

0, 1, 2, 3, 4, 5, . . .

Integers

Whole numbers plus negatives

. . . –4, –3, –2, –1, 0, 1, 2, 3, 4, . . .

Rational Numbers

All numbers of the form , where a and b are integers (but b cannot be zero)

Rational numbers include what we usually call fractions
RESTRICTION: The denominator cannot be zero! (But the numerator can)
If the numerator is zero, then the whole fraction is just equal to zero. If I have zero thirds or zero fourths, than I don’t have anything. However, it makes no sense at all to talk about a fraction measured in “zeroths.”

Fractions can be numbers smaller than 1, like 1/2 or 3/4 (called proper fractions), or they can be numbers bigger than 1 (called improper fractions), like two-and-a-half, which we could also write as 5/2
All integers can also be thought of as rational numbers, with a denominator of 1:

This means that all the previous sets of numbers (natural numbers, whole numbers, and integers) are subsets of the rational numbers.

Now it might seem as though the set of rational numbers would cover every possible case, but that is not so. There are numbers that cannot be expressed as a fraction, and these numbers are called irrational because they are not rational.

Irrational Numbers

Cannot be expressed as a ratio of integers.
As decimals they never repeat or terminate (rationals always do one or the other)
Examples:

3/4 Rational (terminates)

2/3   Rational (repeats)

5/11   Rational (repeats)

5/7 Rational (repeats)

square root of 2: Irrational (never repeats or terminates)

3.1416    Irrational (never repeats or terminates)

The Real Numbers

Rationals + Irrationals
All points on the number line
Or all possible distances on the number line
When we put the irrational numbers together with the rational numbers, we finally have the complete set of real numbers. Any number that represents an amount of something, such as a weight, a volume, or the distance between two points, will always be a real number. The following diagram illustrates the relationships of the sets that make up the real numbers.

#### Tofazzal.ns ##### Re: Number System
« Reply #1 on: May 27, 2015, 06:41:52 PM »
Thanks to share.....
Lecturer, Natural Sciences
Daffodil International University

#### Reza S. H.

• Faculty
• Full Member
• • Posts: 118 ##### Re: Number System
« Reply #2 on: May 28, 2015, 02:27:54 PM »
can you give us the URL?

#### omarsharif

• Full Member
•   • Posts: 162
• Everyday is a good day. ##### Re: Number System
« Reply #3 on: September 22, 2015, 06:24:28 PM »
Complex number is not in the number system ?

#### tanzina_diu

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• • Posts: 165 ##### Re: Number System
« Reply #4 on: March 31, 2016, 11:41:45 AM »
Integers: Whole numbers and the opposites of whole numbers.
Rational Numbers: Numbers that can be written as a fraction (or ratio) of two integers.
Irrational Numbers:  Numbers that cannot be written as a fraction of two integers and are on the number line.
The numbers defined above are in some ways similar as well as different.
The numbers are similar in that they are all components in the real number system.
The numbers are different in their definitions.
Let's look at a few examples...

Integers   Rational   Irrational

5, -9, 0, 2, -16
2/5,  7/9,  3/8,  2.3

Since integers can be written in fractional form they can also be considered rational numbers.  However, not all rational numbers can be integers!
The Real Number System is made up of all the points on the number line. Real Numbers are either integers, rational or irrational. (These are the subsets of the real number system.)

Since integers can be written in fractional form, integers can also be considered rational numbers.

The numbers shown here represent both rational and irrational numbers.
The integer, -2, is a rational number.
The decimal number 0.5 is also a rational number.
Pi ( ) ( 3.14) is a decimal number that, to our knowledge, has no end and cannot be expressed as a ratio of two integers. Therefore it is an irrational number.
Repeating decimals are also rational numbers. Take for example the repeating decimal  . This can be expressed as a ratio of two integers as
Tanzina Hossain
Assistant Professor

#### tanzina_diu

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• • Posts: 165 ##### Re: Number System
« Reply #5 on: March 31, 2016, 11:42:20 AM »
Fundamental Properties of  Real Numbers (R):

The Closure, Identity, Commutative, Associative and Distributive properties of the real numbers are important because they are the fundamental justification for many of the steps taken in algebraic procedures.

Closure

Identity

Commutative

Associative

Distributive   If a and b are real then a+b is real

There exists a real number 0 such that a+0=0+a

If a and b are real then a+b=b+a

If a,b and c are real then (a+b)+c= a(b+c)

If a,b and c are real then a(b+c)=ab+ac
If a and b are real then a.b is real

There exists a real number 1 such that a.1=a

If a and b are real then a.b=b.a

If a,b and c are real then (ab)c= a(bc)

Tanzina Hossain
Assistant Professor