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Assessment 3 ( Computer application ). NAME : SAATHISH ID NO : 2011-1-0011 ( CIE ). NAME : DAARMARAJ ID NO : 2011-01-0007. LECTURER`S NAME : MRS KOH. NEXT. TITTLE. NUMBERS. PREVIOUS. NEXT. Present by. PREVIOUS. NEXT. CONTENT :.

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Assessment 3

( Computer application )

NAME : SAATHISH

ID NO : 2011-1-0011 ( CIE )

NAME : DAARMARAJ

ID NO : 2011-01-0007

LECTURER`S

NAME : MRS KOH

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NUMBERS

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Present by

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CONTENT :

• The Natural Numbers
• The Integers
• The Rational Numbers
• The Irrational Numbers
• The Real Numbers
• Some Properties of Real Numbers
• The RECIPROCAL of Real Number
• Rules for Multiplying Negative Numbers
• Division by Zero

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Real numbers include both rational numbers and irrational numbers. The need for irrational numbers was realized around 500 BC by Greek mathematicians led by Pythagoras . They discovered that the diagonal of square cannot be related to the side as a ratio of two integers, that is, as a rational number; this is the famous discovery that (the square root of 2) is irrational.

Real numbers can be represented as points along an infinitely long number line by adding two elements: + (plus infinity) and - (minus infinity). These elements, however, are not real numbers but are useful in describing various limiting behaviors in calculus and mathematical analysis, especially in measure theory and integration.

The concept of negative numbers was first used around 600 AD in ancient India and China. Negative numbers were not used in Europe until the 17th century , but resistance to the concept persisted for some time. In the 18th century , the Swiss mathematician, Leonard Euler still considered negative solutions to equations as unrealistic.

At the end of the 19th century, Georg Cantor (1845-1918), a German mathematician, introduced set theory to explain the concept of infinity. With this theory, he was able to show that the set of all real numbers is an infinite set.

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The natural numbers are the basic numbers used for counting: 1,2,3,4,4,5,6,….

If we add or multiply any two such numbers we always get another one.

For example,

and

However , if we subtract or divide two natural numbers we don`t always get another natural number.

and

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To overcome the limitation of subtraction , we extend the natural numbers to the system of integers. The integers include the natural numbers, the negative of each natural number, and zero. Thus, we may represent the system of integers by:

If we add, multiply, or subtract any two integers we always get another integer but we still have problems with division.

For example, is not an integer

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The Rational Numbers

To overcome the limitation with division , we extend the system of integers to the system of rational numbers.

This system consists of all the fractions where and are integers with

Note that all the integers are rational. For example,

We can add, multiply, subtract, and divide any two rational numbers and always get another rational number (with division by zero exclude – see the end of this section).

When a rational number is expressed as a decimal, the decimal either terminates or forms a pattern that repeats indefinitely.

For example:

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Irrational Numbers

There exist some numbers in common use that are not rational, i.e. they cannot be expressed as the ratio of two integers. Such numbers are called irrational numbers.

For example: are irrational numbers.

When an irrational number is represented by a decimal, the decimal continues indefinitely without developing any recurrent pattern.

For example:

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THE REAL NUMBERS

• The real numbers consist of all the rationals and all the irrationals combined.
• We can represent the real number on a number line as follows

-4

-3

-2

-1

0

1

2

3

4

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Some Properties Of The Real Numbers

(1)

(2)

(3)

(4)

(5)

(6)

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Example 1:

(a)

Check:

(b)

Check:

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The RECIPROCAL of a real number

If x is any real number except 0,then the reciprocal of x is that real number given by:

Rules for Multiplying Negative Numbers

+ve times +ve = +ve +ve times -ve = -ve

-ve times -ve = +ve -ve times +ve = -ve

That is ,

LIKE times LIKE = + ve

UNLIKES gives – ve.

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Division by Zero (why you can’t)

Note that :

We get two cases (depending on whether 0 or not)

Suppose 0 , and for example 3

Then, , or in other words:

But, 0 for all numbers

And, we know that

So, dividing by zero doesn’t make sense (for the case where )

If ,a slightly different argument is needed

Here, and 0

Then, , or in other words:

But this is true for any choice of

So, can be anything (but we require it to be just one number)

Hence , dividing by zero doesn’t make sense in either case

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Background Music

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SAATHISH

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