## Wednesday, January 18, 2017

### NUMBER SYSTEM

NUMBER SYSTEM AND CLASSIFICATION
=====================================================================

Below is a diagram that shows the classification of the number system (i.e. the different types of numbers in the mathematical world).

Your Task is (1) to define what the following types of numbers and (2) to give some examples.
Please cite the source.
An example on Integer is done for you. Do (2) - (5)
Post your answers as comments as follows
Name (index no.)
answer

(1) Integers

(2) Whole Numbers

(3) Natural Numbers

(4) Rational Numbers

(5) Irrational Numbers

========================================================================
Example:

Mr Johari (008)
(1) Integer
An integer (pronounced IN-tuh-jer) is a whole number (not a fractional number) that can be positive, negative, or zero. Examples of integers are: -5, 1, 5, 8, 97, and 3,043. The set of integers, denoted Z, is formally defined as follows:   Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
source: http://whatis.techtarget.com/

#### 51 comments:

1. This comment has been removed by the author.

2. Hansen (10)

2. Definition
The numbers {0, 1, 2, 3, ...} etc.
There is no fractional or decimal part. And no negatives.
Example: 5, 49 and 980 are all whole numbers.
Source: https://www.mathsisfun.com/definitions/whole-number.html

3. Natural Number
The whole numbers from 1 upwards: 1, 2, 3, and so on ...
Or from 0 upwards in some fields of mathematics: 0, 1, 2, 3 and so on ...
No negative numbers and no fractions.
Source: https://www.mathsisfun.com/definitions/natural-number.html

4. Rational Numbers
A Rational Number is a real number that can be written as a simple fraction (i.e. as a ratio).
Most numbers we use in everyday life are Rational Numbers.
Example:
1.5 is a rational number because 1.5 = 3/2 (it can be written as a fraction)
Source: http://www.mathsisfun.com/rational-numbers.html

5. Irrational Numbers
An Irrational Number is a real number that cannot be written as a simple fraction.
Irrational means not Rational
Example:π = 3.1415926535897932384626433832795 (and more...)
You cannot write down a simple fraction that equals Pi.
Source: https://www.mathsisfun.com/irrational-numbers.html

1. A Distinction effort and quality

3. RATIONAL
A rational number is a number that can be written as a ratio. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers.

The number 8 is a rational number because it can be written as the fraction 8/1.
Likewise, 3/4 is a rational number because it can be written as a fraction.
Even a big, clunky fraction like 7,324,908/56,003,492 is rational, simply because it can be written as a fraction.

Every whole number is a rational number, because any whole number can be written as a fraction. For example, 4 can be written as 4/1, 65 can be written as 65/1, and 3,867 can be written as 3,867/1.

INRATIONAL
All numbers that are not rational are considered irrational. An irrational number can be written as a decimal, but not as a fraction.

An irrational number has endless non-repeating digits to the right of the decimal point. Here are some irrational numbers:

π = 3.141592…

square root of 2 = 1.414213…

Although irrational numbers are not often used in daily life, they do exist on the number line. In fact, between 0 and 1 on the number line, there are an infinite number of irrational numbers!

1. A Distinction effort and quality

4. Ng Zen Haan
A WHOLE NUMBER is a number without fractions; an integer.
Example : 5,49,640
A NATURAL NUMBER is the positive integers (whole numbers) 1, 2, 3, etc., and sometimes zero as well.
A RATIONAL NUMBER is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number. ... A real number that is not rational is called irrational.
A IRRATIONAL NUMBER is a real number that cannot be expressed as a ratio of integers, i.e. as a fraction. Therefore, irrational numbers, when written as decimal numbers, do not terminate, nor do they repeat.
All taken from Google :)

1. A Distinction effort and quality

5. Whole numbers:
Whole Numbers are simply the numbers 0, 1, 2, 3, 4, 5, ... (and so on)
Natural numbers :
"Natural Numbers" can mean either "Counting Numbers" {1, 2, 3, ...}, or "Whole Numbers" {0, 1, 2, 3, ...}, depending on the subject.
Rational numbers
In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number.
Irrational numbers
n mathematics, an irrational number is a real number that cannot be expressed as a ratio of integers, i.e. as a fraction. Therefore, irrational numbers, when written as decimal numbers, do not terminate, nor do they repeat.
Wesley yep

1. Good work but cite source please.

6. Ignatius Lim (014)
(2)Whole Numbers
Whole Number are numbers which do not have a fraction; an integer.(Example:1,2,3,4,5,-1,-2,-3,-4,-5)
Source:Google(No Website Link)
(3)Natural Numbers
Natural Numbers are numbers that are Whole Numbers however not a negative one.(Example:1,2,3,4,5
Source:http://whatis.techtarget.com/definition/natural-number
(4)Rational Numbers
Rational Numbers are numbers which can be turned into a ratio.(Examples:5,1.75,1.5)
Source:http://www.mathsisfun.com/rational-numbers.html
(5)Irrational Numbers are numbers which cannot be turned into a ratio eg. 1 divide 3,the formula pie.
Source:Mr Johari

1. A Distinction effort and quality

7. (2) Whole Numbers
The numbers {0, 1, 2, 3, ...} etc.
There is no fractional or decimal part. And no negatives.
Example: 5, 49 and 980 are all whole numbers.

source: https://www.mathsisfun.com/definitions/whole-number.html

(3) Natural Numbers
The whole numbers from 1 upwards: 1, 2, 3, and so on ...
Or from 0 upwards in some fields of mathematics: 0, 1, 2, 3 and so on ...
No negative numbers and no fractions

source: https://www.mathsisfun.com/definitions/natural-number.html

(4) Rational Numbers
A number that can be made by dividing two integers. (Note: integers have no fractions.)
The word comes from "ratio".

Examples:
• 1/2 is a rational number (1 divided by 2, or the ratio of 1 to 2)
• 0.75 is a rational number (3/4)
• 1 is a rational number (1/1)
• 2 is a rational number (2/1)
• 2.12 is a rational number (212/100)
• −6.6 is a rational number (−66/10)
But Pi is not a rational number, it is an "Irrational Number".

source: https://www.mathsisfun.com/definitions/rational-number.html

(5) Irrational Numbers
An irrational number is simply the opposite of a rational number. A irrational number is one that cannot be represented as the ratio of two integers.
Example - Pi
- √ 2

1. A Distinction effort and quality

2. source for (5) http://www.mathopenref.com/irrational-number.html

3. Love the quality of work shown.

8. (2) Whole Numbers
a whole number is a number without a fraction, decimal and it is not a negative.
e.g. 1, 5, 17, 22.
https://www.mathsisfun.com/definitions/whole-number.html
(3) Natural Numbers
natural numbers are positive integers (no negative numbers and no fractions) counting numbers
e.g. 1,2,3,4,5... onwards.
https://www.mathsisfun.com/definitions/natural-number.html
(4) Rational Numbers
a rational number is a number that can be written as simple fraction (i.e ratio)
e.g 1.5 is a rational number because 1.5 can be written as 3/2
http://www.mathsisfun.com/rational-numbers.html
(5) Irrational Numbers
an irrational number is a real number that cannot be expressed as a ratio of integers, i.e. as a fraction.
e.g. π = 3.14159.... we cannot write down a simple fraction that equals to pi, thus it is an irrational number.
https://www.mathsisfun.com/irrational-numbers.html

1. A Distinction effort and quality

9. Ariel Chia (2)

(2) Whole Numbers
A whole number is a number that has no decimals, fractions, etc. An example of a whole number is 2.

(3) Natural Numbers
A natural number is similar to a whole number. An example of a natural number is 3.

(4) Rational Numbers
A rational number is any number that can be expressed as the quotient or fraction p/q of two integers. An example of a rational number is 8 because it can be written as the fraction 8/1.

(5) Irrational Numbers
An irrational number is a real number that cannot be expressed as a ratio of integers, i.e. as a fraction. It is the opposite of a rational number. An example of an irrational number is π.

source: Wikipedia

1. A Distinction effort and quality

10. whole number- Whole numbers are numbers that has no fraction, an integer
Source: https://www.google.com.sg/webhp?sourceid=chrome-instant&rlz=1C5CHFA_enSG724SG724&ion=1&espv=2&ie=UTF-8#q=whole%20numbers

Natural numbers- The positive integers (whole numbers), and sometimes zero as well
Source: https://www.google.com.sg/webhp?sourceid=chrome-instant&rlz=1C5CHFA_enSG724SG724&ion=1&espv=2&ie=UTF-8#safe=active&q=natural+numbers

Rational numbers- Any number that can be expressed as the quotient or fraction of 2 integers
Source: https://www.google.com.sg/webhp?sourceid=chrome-instant&rlz=1C5CHFA_enSG724SG724&ion=1&espv=2&ie=UTF-8#safe=active&q=what+is+the+meaning+of+rational+number

Irrational number- A real number that cannot be expressed as a ratio of integers, i.e. as a fraction. Therefore, irrational numbers, when written as decimal numbers, do not terminate, nor do they repeat
Source: https://www.google.com.sg/webhp?sourceid=chrome-instant&rlz=1C5CHFA_enSG724SG724&ion=1&espv=2&ie=UTF-8#safe=active&q=what+is+the+meaning+of+irrational+number

1. A Distinction effort and quality

11. (2)whole numbers: a number without fractions; an integer. e.g.:0, 1, 2, 3, 4, 5, ... (and so on)

(3)natural numbers: the positive integers (whole numbers) 1, 2, 3, etc., and sometimes zero as well. 0 upwards in some fields of mathematics: 0, 1, 2, 3 and so on ...
No negative numbers and no fractions.

(4)Rational numbers: In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number. ... A real number that is not rational is called irrational. e.g.• 0.75 is a rational number (3/4)

(5)irrational numbers:In mathematics, an irrational number is a real number that cannot be expressed as a ratio of integers, i.e. as a fraction. Therefore, irrational numbers, when written as decimal numbers, do not terminate, nor do they repeat.e.g. π = 3.1415926535897932384626433832795 (and more...)
You cannot write down a simple fraction that equals Pi.

1. Good work but cite source please.

12. 1)Whole Numbers are simply the numbers 0, 1, 2, 3, 4, 5, ... (and so on)

No Fractions!

Examples: 0, 7, 212 and 1023 are all whole numbers

(But numbers like ½, 1.1 and 3.5 are not whole numbers.)
2)Natural Numbers
"Natural Numbers" can mean either "Counting Numbers" {1, 2, 3, ...}, or "Whole Numbers" {0, 1, 2, 3, ...}, depending on the subject.
counting numbers are Whole Numbers, but without the zero. Because you can't "count" zero.

So they are 1, 2, 3, 4, 5, ... (and so on).
3)n mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number. ... The decimal expansion of an irrational number continues without repeating.Example:

1.5 is a rational number because 1.5 = 3/2 (it can be written as a fraction)

Rational Number
4)In mathematics, an irrational number is a real number that cannot be expressed as a ratio of integers, i.e. as a fraction. Therefore, irrational numbers, when written as decimal numbers, do not terminate, nor do they repeat.Example: π (Pi) is a famous irrational number. You cannot write down a simple fraction that equals Pi. The popular approximation of 22/7 = 3.1428571428571... is close but not accurate. Another clue is that the decimal goes on forever without repeating.

sources: wikipedia and https://www.mathsisfun.com/whole-numbers.html

1. A Distinction effort and quality

13. Audrey (05)

Whole numbers are the numbers 1,2,3,4,5,6,7,8,9 and so on. Whole numbers do not have decimals

Natural numbers care positive integers. It can mean counting numbers or whole numbers depending on the subject. Natural numbers also do not have decimals

Rational numbers are numbers that can be written as a simple fraction or ratio (e.g. 1.5)
1.5 is a rational number because 1.5 = 3/2 (it can be written as a fraction)

An Irrational Number is a real number that cannot be written as a simple fraction. Pi is an irrational number because it cannot be written down as a simple fraction although we often estimate it as 22/7 or 3.14. those are close but not accurate.

1. Source: ttps://www.mathsisfun.com

2. A Distinction effort and quality

14. 2. Whole numbers
The numbers {0, 1, 2, 3, ...} etc.
There is no fractional or decimal part. And no negatives.

https://www.mathsisfun.com/definitions/whole-number.html

3.Natural Numbers
No negative numbers and no fractions.

https://www.mathsisfun.com/definitions/natural-number.html

4.Rational Numbers
A Rational Number is a real number that can be written as a simple fraction (i.e. as a ratio).

http://www.mathsisfun.com/rational-numbers.html

5. Irrational Numbers
An Irrational Number is a real number that cannot be written as a simple fraction.

https://www.mathsisfun.com/irrational-numbers.html

1. A Distinction effort and quality

15. 1)A natural number is a number that occurs commonly and obviously in nature. As such, it is a whole, non-negative number. The set of natural numbers, denoted N, can be defined in either of two ways:

N = {0, 1, 2, 3, ...}

N = (1, 2, 3, 4, ...}

2)A whole number is fractional or decimal part. And no negatives.The numbers {0, 1, 2, 3, ...} etc
3)A rational number is a number that can be written as a ratio. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers. The number 8 is a rational number because it can be written as the fraction 8/1.
4)In mathematics, an irrational number is a real number that cannot be expressed as a ratio of integers, i.e. as a fraction. Therefore, irrational numbers, when written as decimal numbers, do not terminate, nor do they repeat.
Shaun Ho (19)

1. Good but cite source please.

16. Joel Tan(22)
(4)Rational numbers
A rational number is a number that can be written as a ratio. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers.Every whole number is a rational number, because any whole number can be written as a fraction. Examples:The number 8 is a rational number because it can be written as the fraction 8/1.
Likewise, 3/4 is a rational number because it can be written as a fraction.
Even a big, clunky fraction like 7,324,908/56,003,492 is rational, simply because it can be written as a fraction.

sources: http://www.factmonster.com/ipka/A0876704.html

1. good but where are the other numbers?

17. How Hong Jie (11)
(2) Whole Numbers
Also called counting number. one of the positive integers or zero; any of the numbers (0, 1, 2, 3,
…).
source: dictionary.com

(3)Natural Numbers
In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country"). In common language, words used for counting are "cardinal numbers" and words used for ordering are "ordinal numbers"The natural numbers are the basis from which many other number sets may be built by extension: the integers, by including an additive inverse (−n) for each natural number n (and zero, if it is not there already, as its own additive inverse); the rational numbers, by including a multiplicative inverse (1/n) for each nonzero integer n; the real numbers by including with the rationals the (converging) Cauchy sequences of rationals; the complex numbers, by including with the real numbers the unresolved square root of minus one; and so on.[6][7] These chains of extensions make the natural numbers canonically embedded (identified) in the other number systems.
Source:en.wikipedia.org

(4)Rational Numbers:
In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.[1] Since q may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to as "the rationals", is usually denoted by a boldface Q (or blackboard bold
Q
\mathbb {Q} , Unicode ℚ);[2] it was thus denoted in 1895 by Giuseppe Peano after quoziente, Italian for "quotient".
Source: en.wikipedia.org

(5)Irrational numbers:
In mathematics, an irrational number is a real number that cannot be expressed as a ratio of integers, i.e. as a fraction. Therefore, irrational numbers, when written as decimal numbers, do not terminate, nor do they repeat. For example, the number π starts with 3.14159265358979, but no finite number of digits can represent it exactly and it does not end in a segment that repeats itself infinitely often. The same can be said for any irrational number.

As a consequence of Cantor's proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational.[1]

When the ratio of lengths of two line segments is irrational, the line segments are also described as being incommensurable, meaning they share no measure in common.

Numbers that are irrational include the ratio π of a circle's circumference to its diameter, Euler's number e, the golden ratio φ, and the square root of two;[2][3][4] in fact all square roots of natural numbers, other than of perfect squares, are irrational.
Source: en.wikipedia.org

1. A Distinction effort and quality - appreciate it!

18. 2.Also called counting number. one of the positive integers or zero; any of the numbers (0, 1, 2, 3, …).

3. In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country"). In common language, words used for counting are "cardinal numbers" and words used for ordering are "ordinal numbers".

4.In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.[1] Since q may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to as "the rationals",

5.In mathematics, an irrational number is a real number that cannot be expressed as a ratio of integers, i.e. as a fraction. Therefore, irrational numbers, when written as decimal numbers, do not terminate, nor do they repeat. For example, the number π starts with 3.14159265358979, but no finite number of digits can represent it exactly and it does not end in a segment that repeats itself infinitely often. The same can be said for any irrational number.

1. Good but cite source please.

19. Name: Yeo Teng Jun
Class: S107
Index: 24

2. Whole Numbers
Whole numbers are a set of numbers which are not negative integers and excludes fractions which range from 0 to infinity.
Example: 5 is a whole number as it is a number that is not a fraction.

3. Natural Numbers
Natural Numbers are used for counting, which ranges from 1 to infinity. A natural number is a number that occurs commonly and obviously in nature. As such, it is a whole, non-negative number.
Example: 5 is a natural numbers as it is most commonly used for daily life.

Source: https://en.wikipedia.org/wiki/Natural_number

4. Rational Numbers
A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number.
Example: 1/2 is a rational number as it is a fraction.

Source: https://en.wikipedia.org/wiki/Rational_number

5. Irrational Numbers
Irrational numbers are integers that has never ending digits.
Example: Pi is an irrational number cause Pi = 3.1415926535897932384626446........ and so on.
1/3 is an irrational number as 1/3 is = 0.3333333333333333333333333333....and so on.

1. A Distinction effort and quality

20. Silas Benaiah Lew ji hin(20)
1)An integer (pronounced IN-tuh-jer) is a whole number (not a fractional number) that can be positive, negative, or zero. Examples of integers are: -5, 1, 5, 8, 97, and 3,043. Examples of numbers that are not integers are: -1.43, 1 3/4, 3.14, .09, and 5,643.1.
source:Wikepedia
2)whole numbers are numbers with no fractions; an integer.E.G.1 5 678
source:wikepedia
3)the positive integers (whole numbers) 1, 2, 3, etc., and sometimes zero a.E.g. 1 2 3 4 5 6 7 8 9 10
source:wikepedia
4)In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number. ... The decimal expansion of an irrational number continues without repeating.e.g. 0.4,0.75,5,4
source:wikepedia
5)In mathematics, an irrational number is a real number that cannot be expressed as a ratio of integers, i.e. as a fraction. Therefore, irrational numbers, when written as decimal numbers, do not terminate, nor do they repeat.e.g 0.333333333333333333,33.33333333
source Wikipedia

1. A Distinction effort and quality

21. Timothy Luk (23)
irrational numbers:
an irrational number is a real number that cannot be expressed as a ratio of integers, i.e. as a fraction. Therefore, irrational numbers, when written as decimal numbers, do not terminate, nor do they repeat. pi is a good example of an irrational number
π=3.141592653589793238462643383279502884197169399327980...etc
source :wikipedia
rational number:
A rational number is a number that can be written as a ratio. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers.The number 8 is a rational number because it can be written as the fraction 8/1.
source: fact monster

1. A Distinction effort and quality

22. rational numbers:
In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.[1] Since q may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to as "the rationals",

23. 2.Also called counting number. one of the positive integers or zero; any of the numbers (0, 1, 2, 3, …).

3. In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country"). In common language, words used for counting are "cardinal numbers" and words used for ordering are "ordinal numbers".

4.In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.[1] Since q may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to as "the rationals",

5.In mathematics, an irrational number is a real number that cannot be expressed as a ratio of integers, i.e. as a fraction. Therefore, irrational numbers, when written as decimal numbers, do not terminate, nor do they repeat. For example, the number π starts with 3.14159265358979, but no finite number of digits can represent it exactly and it does not end in a segment that repeats itself infinitely often. The same can be said for any irrational number.

1. good but cite source please.

24. Ryan Lee(13)
(2)Whole numbers
Whole numbers are like integers,just that it does not include negative numbers
Examples are 0,1,2,3
(3)Natural numbers
natural numbers are positive integers
Examples are 8,0,
(4)Rational numbers
Rational numbers are numbers that can be expressed as a fraction
Examples are 7/11
(5)Irrational numbers
Irrational numbers are numbers that cannot be expressed as a fraction
Examples are pi, root 2,root 3 and root 5

1. good but cite source please.

25. Chua Xing Zi (3)
(2) whole numbers
Whole numbers are numbers without fractions and decimals. There is no negatives. All whole numbers are positive.Some examples of whole numbers are 5,49 and 980.

(3)Natural Numbers
A natural number is a number that occurs commonly and obviously in nature. As such, it is a whole, non-negative number.Some natural numbers are 1,2,3,4, and 5.

(4)Rational numbers
A rational number is a real number that can be written as a simple fraction or as a ratio. Most numbers we use in everyday life are rational numbers.Some examples of a rational number is 1.5,1.75 or even 5.

(5)Irrational numbers
An irrational number is a real number that cannot be expressed as a ratio or an integer .

1. good but cite source please.

Note: Only a member of this blog may post a comment.