Rational Numbers - Grade 10 [CAPS]

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1 OpenStx-CNX module: m848 Rtionl Numers - Grde 0 [CAPS] Free High School Science Texts Project Bsed on Rtionl Numers y Rory Adms Free High School Science Texts Project Mrk Horner Hether Willims This work is produced y OpenStx-CNX nd licensed under the Cretive Commons Attriution License.0 Introduction As descried in Review of pst work, numer is wy of representing quntity. The numers tht will e used in high school re ll rel numers, ut there re mny dierent wys of writing ny single rel numer. This chpter descries rtionl numers. Khn Acdemy video on Integers nd Rtionl Numers This medi oject is Flsh oject. Plese view or downlod it t < Figure Version.4: Jun, 0 9:5 pm "Review of Pst Work - Grde 0 [CAPS]" <

2 OpenStx-CNX module: m848 The Big Picture of Numers Figure The term whole numer does not hve consistent denition. Vrious uthors use it in mny dierent wys. We use the following denitions: nturl numers re (,,,...) whole numers re (0,,,,...) integers re (... -, -, -, 0,,,,...) Denition The following numers re ll rtionl numers. 0, 7,, 0 0, 6 You cn see tht ll denomintors nd ll numertors re integers. Denition : Rtionl Numer A rtionl numer is ny numer which cn e written s: where nd re integers nd 0. Note tht ecuse we cn write (in other words, one cn lwys nd n equivlent rtionl expression where > 0) mthemticins typiclly dene rtionl numers not s oth nd eing integers, ut rther tht is n integer nd is nturl numer. This voids hving to worry out zero in the denomintor. s tip: Only frctions which hve numertor nd denomintor (tht is not 0) tht re integers re rtionl numers. This mens tht ll integers re rtionl numers, ecuse they cn e written with denomintor of. Therefore 7, π () 0 re not exmples of rtionl numers, ecuse in ech cse, either the numertor or the denomintor is not n integer. A numer my not e written s n integer divided y nother integer, ut my still e rtionl numer. This is ecuse the results my e expressed s n integer divided y n integer. The rule is, if numer cn e written s frction of integers, it is rtionl even if it cn lso e written in nother wy s well. Here re two exmples tht might not look like rtionl numers t rst glnce ut re ecuse there re equivlent forms tht re expressed s n integer divided y nother integer: () (), = 00, 6, 9 = = 00 (4)

3 OpenStx-CNX module: m848. Rtionl Numers. If is n integer, is n integer nd c is irrtionl, which of the following re rtionl numers? c. d. c Click here for the solution. If is rtionl numer, which of the following re vlid vlues for?.. 0 c. d., Click here for the solution 4 Forms of Rtionl Numers All integers nd frctions with integer numertors nd denomintors re rtionl numers. There re two more forms of rtionl numers. 4. Investigtion : Deciml Numers You cn write the rtionl numer s the deciml numer 0,5. Write the following numers s decimls: Do the numers fter the deciml comm end or do they continue? If they continue, is there repeting pttern to the numers? You cn write rtionl numer s deciml numer. Two types of deciml numers cn e written s rtionl numers:. deciml numers tht end or terminte, for exmple the frction 4 0 cn e written s 0,4.. deciml numers tht hve repeting pttern of numers, for exmple the frction cn e written s 0,. The dot represents recurring 's i.e., 0,... = 0,. For exmple, the rtionl numer 5 6 cn e written in deciml nottion s 0, 8 nd similrly, the deciml numer 0,5 cn e written s rtionl numer s 4. tip: You cn use r over the repeted numers to indicte tht the deciml is repeting deciml

4 OpenStx-CNX module: m Converting Terminting Decimls into Rtionl Numers A deciml numer hs n integer prt nd frctionl prt. For exmple 0, 589 hs n integer prt of 0 nd frctionl prt of 0, 589 ecuse 0 + 0, 589 = 0, 589. The frctionl prt cn e written s rtionl numer, i.e. with numertor nd denomintor tht re integers. Ech digit fter the deciml point is frction with denomintor in incresing powers of ten. For exmple: 0 00 is 0, is 0, 0 This mens tht: 0, 589 = = = (5) 5. Frctions. Write the following s frctions:. 0,. 0, c. 0, 58 d. 0, 589 Click here for the solution 4 6 Converting Repeting Decimls into Rtionl Numers When the deciml is repeting deciml, it more work is needed to write the frctionl prt of the deciml numer s frction. We will explin y mens of n exmple. If we wish to write 0, in the form (where nd re integers) then we would proceed s follows x = 0,... 0x =,... multiply y 0 on oth sides 9x = (sutrcting the second eqution from the rst eqution) x = 9 = And nother exmple would e to write 5, 4 s rtionl frction. x = 5, x = 54, multiply y 000 on oth sides 999x = 547 (sutrcting the second eqution from the rst eqution) 547 x = 999 = 0 7 (6) (7) 4

5 OpenStx-CNX module: m848 5 For the rst exmple, the deciml ws multiplied y 0 nd for the second exmple, the deciml ws multiplied y 000. This is ecuse for the rst exmple there ws only one digit (i.e. ) recurring, while for the second exmple there were three digits (i.e. 4) recurring. In generl, if you hve one digit recurring, then multiply y 0. If you hve two digits recurring, then multiply y 00. If you hve three digits recurring, then multiply y 000. Cn you spot the pttern yet? The numer of zeros is the sme s the numer of recurring digits. Not ll deciml numers cn e written s rtionl numers. Why? Irrtionl deciml numers like =, cnnot e written with n integer numertor nd denomintor, ecuse they do not hve pttern of recurring digits. However, when possile, you should try to use rtionl numers or frctions insted of decimls. 6. Repeted Deciml Nottion. Write the following using the repeted deciml nottion:. 0,.... 0,... c. 0,... d. 0, Click here for the solution 5. Write the following in deciml form, using the repeted deciml nottion:.. c d. 9 Click here for the solution 6. Write the following decimls in frctionl form:. 0, 6. 5, c. 0, Click here for the solution 7 7 Summry ˆ Rel numers cn e either rtionl or irrtionl. ˆ A rtionl numer is ny numer which cn e written s ˆ The following re rtionl numers:. Frctions with oth denomintor nd numertor s integers.. Integers. c. Deciml numers tht end. d. Deciml numers tht repet. where nd re integers nd

6 OpenStx-CNX module: m End of Chpter Exercises. If is n integer, is n integer nd c is irrtionl, which of the following re rtionl numers? c. d. c Click here for the solution 8. Write ech deciml s simple frction:. 0, 5. 0, c. 0, 6 d., 59 e., 7 7 Click here for the solution 9. Show tht the deciml, 8 is rtionl numer. Click here for the solution 0 4. Express 0, 7 8 s frction where, Z (show ll working). Click here for the solution