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Inleiding

'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, ....)

Definisie

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 ˙ .

Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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