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Grade 9


Module 1

Numbers – where do they come from?

Numbers – where do they come from?


1 Our name for the set of Natural numbers is N, and we write it: N = { 1 ; 2 ; 3 ; . . . }

1.1 Will the answer always be a natural number if you add any two natural numbers? How will you convince someone that it is always the case?

1.2 Multiply any two natural numbers. Is the answer always also a natural number?

1.3 Now subtract any natural number from any other natural number. Describe all the sorts of answers you can expect. Try to write down why this happens.

2 To deal with the answers you got in 1.3, we have to extend the number system to include zero and negative numbers – we call them, with the natural numbers, the integers. They are called Z and this is one way to write them down: Z = { 0 ; ±1 ; ±2 ; ±3 ; . . . }

2.1 Complete the following definitions by writing down what has to be inside the brackets:

  • Counting numbers N 0 = {.........................}
  • Integers Z = {.........................} in another way! (Integers are also called whole numbers)

3 Is the answer always another integer when you divide any integer by any other integer (except zero)?To allow for these answers we have to extend the number system to the rational numbers:

3.1 Q (rational numbers) is the set of all the numbers which can be written in the form a b size 12{ { {a} over {b} } } {} where a and b are integers as long as b is not zero. Explain very clearly why b is not allowed to be zero.

4 Q` (irrational numbers) is the set of numbers which cannot be written as a common fraction, and are therefore not in Q. Putting Q en Q` together gives the set called R, the real numbers.

4.1 Write down what you think is in the set R` . They are called non-real numbers.


Quipu is an Inca word meaning a string (or set of strings) with knots in it. This system was used for remembering things, mainly numbers. It was used widely in the ancient world; not only in South America. At its simplest, it was just one string with each knot representing one item. In more advanced systems, more strings were used, often of different colours; sometimes a system of place-values was used.


1. What is the importance of having a symbol for zero? Think about all the things we’ll be unable to do if we didn’t have a zero.

2 Find out what we call the set of numbers we get when putting R and R` together. Can you say more about them?

3 Design your own set of number symbols like those in table 1. Show how any number can be written in your system. Now think up new symbols for + and – and × and , and then make up a few sums to show how your system works.



Let’s check out the rational numbers

  • Do the following sums on your own calculator to confirm that they are correct:
  • Remember to do the operations in the proper order.
  1. 2 + 3  100 + 1 + 1  10 = 3,013

Is 3,013 a rational number? Yes! Look at this bit of magic:

3,013 = 3 1 + 13 1000 size 12{ { {3} over {1} } + { {"13"} over {"1000"} } } {} = 3000 1000 + 13 1000 size 12{ { {"3000"} over {"1000"} } + { {"13"} over {"1000"} } } {} = 3000 + 13 1000 size 12{ { {"3000"+"13"} over {"1000"} } } {} = 3013 1000 size 12{ { {"3013"} over {"1000"} } } {}

It is easy to write it down straightaway. Explain the method carefully.

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Mathematics grade 9. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col11056/1.1
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