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NAME : SAATHISH ID NO : 2011-1-0011 ( CIE )

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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NAME : SAATHISH ID NO : 2011-1-0011 ( CIE )

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  1. 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

  2. TITTLE NUMBERS PREVIOUS NEXT

  3. Present by PREVIOUS NEXT

  4. 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 PREVIOUS NEXT

  5. E B S R M U N 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. PREVIOUS NEXT

  6. T H E N A T U R A L N U M B E R 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 PREVIOUS NEXT

  7. T H E I N T E G E R S 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 PREVIOUS NEXT

  8. 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: PREVIOUS NEXT

  9. 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: PREVIOUS NEXT

  10. 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 PREVIOUS NEXT

  11. Some Properties Of The Real Numbers (1) (2) (3) (4) (5) (6) PREVIOUS NEXT

  12. Example 1: (a) Check: (b) Check: PREVIOUS NEXT

  13. 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. PREVIOUS NEXT

  14. 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 PREVIOUS NEXT

  15. R E F E R E N C E Best animation PowerPoint Templates Background Music PREVIOUS NEXT

  16. SAATHISH AND Raj PREVIOUS

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