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Enrichment
Emphasize that the integers provides closure for the operation of subtraction and also allows subtraction to be defined in terms of the addition of integers.
Show that the whole numbers are a proper subset of the integers and that the counting numbers are a proper subset of the whole numbers.
Ask the student to consider whether the set of whole numbers is larger (has more members) than the counting numbers. Is the set of integers larger than the set of whole numbers. Show how we count the sets by one-one correspondence. Introduce the infinity symbol as the measure of an unbounded set.
Ask the student to consider whether there is a set of numbers which is closed under the operation of division. This is the set of rational numbers. What restriction would have to be made for this to be possible (p/q where q is not zero). This derives from the property that if a*b = 0 either a=0 or b=0.
Ask the student whether the set of rational numbers is complete. Point out that there are points on a number line which are not rational (e.g. square root of 2). Show the proof that the square root of 2 is not rational -
p/q * p/q = 2 (where p and q are integers, q not 0)
p * 1/q * p * 1/q = 2
p*p = 2 q * q
therefore p = 2*t and p * p = 2*2*t*t = 2*q*q and 2*t*t = q*q therefore q is even
But p and q were relatively prime.
Why would the square root of two be called an irrational number?
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