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'n Getal (soos beskryf in die hersieningshoofstuk) is 'n manier om 'n hoeveelheid voor te stel. Die getalle wat op hoërskool gebruik sal word is almal reëel, maar daar is heelwat verskillende maniere om enige gegewe reële getal voor te stel.

Hierdie hoofstuk beskryf rasionale getalle .

Khan academy video oor heelgetalle en rasionale getalle (in engels)

Die oorhoofse beskouing van getalle

Die term "heelgetal" het nie 'n konsekwente definisie nie. Verskillende skrywers gebruik dit op verskillende wyses. Ons gebruik die volgende definisies:

  • natuurlike getalle is (1, 2, 3, ...)
  • telgetalle is (0, 1, 2, 3, ...)
  • heelgetalle is (... -3, -2, -1, 0, 1, 2, 3, ....)


Die volgende getalle is almal rasionaal

10 1 , 21 7 , - 1 - 3 , 10 20 , - 3 6

Jy kan sien dat al die tellers en noemers heelgetalle is.

Rasionale getal

'n Rasionale getal is enige getal wat geskryf kan word as:

a b

waar a en b heelgetalle is en b 0 .

Slegs breuke wat 'n heeltallige teller en noemer het (wat nie 0 is nie), is rasionale getalle.

Dit beteken dat alle heelgetalle rasionaal is, aangesien hulle geskryf kan word met 'n noemer van 1.

Dus is

2 7 , π 20

nie voorbeelde van rasionale getalle nie, want in elke geval is óf die teller óf die noemer nie 'n heelgetal nie.

'n Getal wat nie geskryf word in die vorm van 'n heelgetal gedeel deur 'n heelgetal nie kan nogtans 'n rasionale getal wees. Dit is omdat die vereenvoudigde resultaat wel as 'n kwosiënt van heelgetalle geskryf kan word. Die reël is dat indien 'n getal geskryf kan word as 'n kwosiënt van heelgetalle, dit rasionaal is, selfs al kan dit op 'n manier geskryf word wat nie so 'n kwosiënt is nie. Hier is twee voorbeelde wat dalk nie na rasionale getalle lyk nie, maar nogtans is, omdat daar ekwivalente vorms gevind kan word wat bestaan uit 'n heelgetal gedeel deur 'n heelgetal:

- 1 , 33 - 3 = 133 300 , - 3 6 , 39 = - 300 639 = - 100 213

Rasionale getalle

  1. Indien a 'n heelgetal is, b 'n heelgetal is en c irrasionaal is, watter van die volgende is rasionale getalle?
    (i) 5 6 (ii) a 3 (iii) b 2 (iv) 1 c
  2. Indien a 1 'n rasionale getal is, watter van die volgende is geldige waardes vir a ?
    (i) 1 (ii) - 10 (iii) 2 (iv) 2 , 1

Vorme van rasionale getalle

Alle heelgetalle en heeltallige kwosiënte is rasionaal. Daar is twee bykomende vorme van rasionale getalle.

Ondersoek: desimale getalle

Jy kan die rasionale getal 1 2 skryf as die desimale getal 0,5. Skryf die volgende getalle as desimale getalle:

  1. 1 4
  2. 1 10
  3. 2 5
  4. 1 100
  5. 2 3

Beskou die getalle na die desimale komma. Kom hulle tot 'n einde of gaan hulle voort? Indien hulle voortgaan, is daar 'n herhalende patroon in die getalle?

Jy kan 'n rasionale getal as 'n desimale getal skryf. Twee tipes desimale getalle wat as rasionale getalle geskryf kan word:

  1. Desimale getalle waarvan die nie-nul getalle na die komma tot 'n einde kom of termineer , byvoorbeeld die breuk 4 10 kan geskryf word as 0,4.
  2. Desimale getalle wat 'n nimmereindigende herhalende patroon van getalle na die komma het, byvoorbeeld die breuk 1 3 kan geskryf word as 0 , 3 ˙ . Die dot beteken dat die 3 'e repeteer, m.a.w. 0 , 333 ... = 0 , 3 ˙ .

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Source:  OpenStax, Siyavula textbooks: wiskunde (graad 10) [caps]. OpenStax CNX. Aug 04, 2011 Download for free at http://cnx.org/content/col11328/1.4
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