9th-RATIONAL NUMBERS

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Published on May 31, 2013

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PowerPoint Presentation: 5 7 2 1 Rational Numbers S.JASMINE SUGIRTHA 9 th CLASS S.JASMINE SUGIRTHA 1 MATHS PowerPoint Presentation: Rational and Irrational Numbers Rational Numbers A rational number is any number that can be expressed as the ratio of two integers. Examples All terminating and repeating decimals can be expressed in this way so they are irrational numbers. a b 4 5 2 2 3 = 8 3 6 = 6 1 2.7 = 27 10 0.625 = 5 8 34.56 = 3456 100 -3 = 3 1 - 0.3 = 1 3 0.27 = 3 11 0.142857 = 1 7 0.7 = 7 10 S.JASMINE SUGIRTHA 2 MATHS Rational: Rational and Irrational Numbers Rational Numbers A rational number is any number that can be expressed as the ratio of two integers. All terminating and repeating decimals can be expressed in this way so they are irrational numbers. a b Show that the terminating decimals below are rational. 0.6 3.8 56.1 3.45 2.157 6 10 38 10 561 10 345 100 2157 1000 Rational S.JASMINE SUGIRTHA 3 MATHS PowerPoint Presentation: Rational and Irrational Numbers Rational Numbers A rational number is any number that can be expressed as the ratio of two integers. All terminating and repeating decimals can be expressed in this way so they are irrational numbers. a b To show that a repeating decimal is rational. Example 1 To show that 0.333… is rational. Let x = 0.333… 10 x = 3.33… 9 x = 3 x = 3/9 x = 1/3 Example 2 To show that 0.4545… is rational. Let x = 0.4545… 100 x = 45.45… 99 x = 45 x = 45/99 x = 5/11 S.JASMINE SUGIRTHA 4 MATHS PowerPoint Presentation: Rational and Irrational Numbers Rational Numbers A rational number is any number that can be expressed as the ratio of two integers. All terminating and repeating decimals can be expressed in this way so they are irrational numbers. a b Question 1 Show that 0.222… is rational. Let x = 0.222… 10 x = 2.22… 9 x = 2 x = 2/9 Question 2 Show that 0.6363… is rational. Let x = 0.6363… 100 x = 63.63… 99 x = 63 x = 63/99 x = 7/11 S.JASMINE SUGIRTHA 5 MATHS PowerPoint Presentation: Rational and Irrational Numbers Rational Numbers A rational number is any number that can be expressed as the ratio of two integers. All terminating and repeating decimals can be expressed in this way so they are irrational numbers. a b 999 x = 273 x = 273/999 9999 x = 1234 x = 1234/9999 Question 3 Show that 0.273is rational. Let x = 0.273 1000 x = 273.273 x = 91/333 Question 4 Show that 0.1234 is rational. Let x = 0.1234 10000 x = 1234.1234 S.JASMINE SUGIRTHA 6 MATHS PowerPoint Presentation: Rational and Irrational Numbers Rational Numbers A rational number is any number that can be expressed as the ratio of two integers. All terminating and repeating decimals can be expressed in this way so they are irrational numbers. a b By looking at the previous examples can you spot a quick method of determining the rational number for any given repeating decimal. 0.1234 1234 9999 0.273 273 999 0.45 45 99 0.3 3 9 S.JASMINE SUGIRTHA 7 MATHS PowerPoint Presentation: Rational and Irrational Numbers Rational Numbers A rational number is any number that can be expressed as the ratio of two integers. All terminating and repeating decimals can be expressed in this way so they are irrational numbers. a b 0.1234 1234 9999 0.273 273 999 0.45 45 99 0.3 3 9 Write the repeating part of the decimal as the numerator and write the denominator as a sequence of 9’s with the same number of digits as the numerator then simplify where necessary. S.JASMINE SUGIRTHA 8 MATHS PowerPoint Presentation: Rational and Irrational Numbers Rational Numbers A rational number is any number that can be expressed as the ratio of two integers. All terminating and repeating decimals can be expressed in this way so they are irrational numbers. a b 1543 9999 628 999 32 99 7 9 0.1543 0.628 0.32 0.7 Write down the rational form for each of the repeating decimals below. S.JASMINE SUGIRTHA 9 MATHS Irrational: Irrational a b Rational and Irrational Numbers Irrational Numbers An irrational number is any number that cannot be expressed as the ratio of two integers. 1 1 Pythagoras The history of irrational numbers begins with a discovery by the Pythagorean School in ancient Greece. A member of the school discovered that the diagonal of a unit square could not be expressed as the ratio of any two whole numbers. The motto of the school was “All is Number” (by which they meant whole numbers). Pythagoras believed in the absoluteness of whole numbers and could not accept the discovery. The member of the group that made it was Hippasus and he was sentenced to death by drowning. S.JASMINE SUGIRTHA 10 MATHS PowerPoint Presentation: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Rational Numbers Irrational Numbers S.JASMINE SUGIRTHA 11 MATHS PowerPoint Presentation: a b Rational and Irrational Numbers Irrational Numbers An irrational number is any number that cannot be expressed as the ratio of two integers. 1 1 Pythagoras Intuition alone may convince you that all points on the “Real Number” line can be constructed from just the infinite set of rational numbers, after all between any two rational numbers we can always find another. It took mathematicians hundreds of years to show that the majority of Real Numbers are in fact irrational. The rationals and irrationals are needed together in order to complete the continuum that is the set of “Real Numbers”. S.JASMINE SUGIRTHA 12 MATHS PowerPoint Presentation: a b Rational and Irrational Numbers Irrational Numbers An irrational number is any number that cannot be expressed as the ratio of two integers. 1 1 Pythagoras Surds are Irrational Numbers We can simplify numbers such as into rational numbers. However, other numbers involving roots such as those shown cannot be reduced to a rational form. Any number of the form which cannot be written as a rational number is called a surd. All irrational numbers are non-terminating , non-repeating decimals. Their decimal expansion form shows no pattern whatsoever. Other irrational numbers include and e , ( Euler’s number ) S.JASMINE SUGIRTHA 13 MATHS PowerPoint Presentation: Rational and Irrational Numbers Multiplication and division of surds. For example: and also for example and S.JASMINE SUGIRTHA 14 MATHS PowerPoint Presentation: Rational and Irrational Numbers Example questions Show that is rational rational Show that is rational rational a b S.JASMINE SUGIRTHA 15 MATHS PowerPoint Presentation: Rational and Irrational Numbers Questions a e State whether each of the following are rational or irrational. b c d f g h irrational rational rational irrational rational rational rational irrational S.JASMINE SUGIRTHA 16 MATHS PowerPoint Presentation: Rational and Irrational Numbers Combining Rationals and Irrationals Addition and subtraction of an integer to an irrational number gives another irrational number, as does multiplication and division. Examples of irrationals S.JASMINE SUGIRTHA 17 MATHS PowerPoint Presentation: Rational and Irrational Numbers Combining Rationals and Irrationals Multiplication and division of an irrational number by another irrational can often lead to a rational number. Examples of Rationals 21 26 8 1 -13 S.JASMINE SUGIRTHA 18 MATHS PowerPoint Presentation: Rational and Irrational Numbers Combining Rationals and Irrationals Determine whether the following are rational or irrational. (a) 0.73 (b) (c) 0.666…. (d) 3.142 (e) (f) (g) (h) (i) (j) (j) (k) (l) irrational rational rational rational irrational irrational irrational rational rational irrational irrational rational rational S.JASMINE SUGIRTHA 19 MATHS END: END THANK U S.JASMINE SUGIRTHA 20 MATHS

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