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Numbers – where do they come from?
CLASS WORK
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:
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 $\frac{a}{b}$ 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.
end of CLASS WORK
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.
HOMEWORK ASSIGNMENT
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.
end of HOMEWORK ASSIGNMENT
ENRICHMENT ASSIGNMENT
Let’s check out the rational numbers
Is 3,013 a rational number? Yes! Look at this bit of magic:
3,013 = $\frac{3}{1}+\frac{\text{13}}{\text{1000}}$ = $\frac{\text{3000}}{\text{1000}}+\frac{\text{13}}{\text{1000}}$ = $\frac{\text{3000}+\text{13}}{\text{1000}}$ = $\frac{\text{3013}}{\text{1000}}$
It is easy to write it down straightaway. Explain the method carefully.
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