Structures 1

1. Structures 1 Number Systems

2. What is a number? • How might you describe the set of all numbers? • How do you visualise the set of all numbers?

3. Natural numbers (N) • “counting numbers” • 1, 2, 3, 4, … Whole numbers = N + {0} • 0, 1, 2, 3, …

4. what can you do within the set of natural numbers? • addition? • subtraction? • multiplication? • division? • solve the equations

5. Integers (Z) • positive and negative and zero • closure under addition and subtraction • additive inverse and identity • solution to all equations of the form

6. But … • Negative numbers are difficult and have only relatively recently been accepted in the West. • see article by Jill Howard athttp://nrich.maths.org/5747

7. 200 BC - Chinese number rods • red – positive • black – negative • Used in commercial and tax calculations • black and red cancel each other out

8. Diophantus (200-c.284 AD) called the result of the equation 4=4x+20 “absurd” • Al-Khwarizmi (c.780 – c.850 AD) – considered as an originator of algebra, but treated negative results as meaningless • Arabic mathematicians from the 10th century began to use and accept negative results

9. In Europe … • From the 15th century, negative numbers started to be used, initially in commercial applications and then more generally in equations and calculus • But still with discomfort about their meaning

10. They are useful only, in so far as I am able to judge, to darken the very whole doctrines of the equations and to make dark of the things that are in their nature excessively obvious and simple. It would have been desirable in consequence that the negative roots were never allowed in algebra or that they were discarded. Francis Maseres (1759)

11. What difficulties do pupils have and how can you support them to make meanings for negative numbers?

12. Representations of positive and negative integers • temperature • bank balance • number line • lifts/ hot air balloons • yin and yang • … • What are these representations, models and metaphors helpful for and where do they fail?

13. -1+3=+2 +3 signed numbers are POSITIONS and MOVEMENTS on the number line -4 -3 -1 +1 -2 0 +2 +3 +4 -3 -1-3=-4

14. -1+3=+2 +3 signed numbers are POSITIONS and MOVEMENTS on the number line -4 -3 -1 +1 -2 0 +2 +3 +4 -3 -1-3=-4

15. Why do two negatives make a positive? (And what does this mean?)

16. Negative numbers as part of a coherent system • all equations of the form x+a=b have a solution • patterns continue consistently • proof based on rules of arithmetic

17. Using a familiar pattern

18. anything times zero …

19. anything times one … (identity)

20. complete the pattern …

21. complete the pattern …

22. A proof using algebra Define a number x as: Then And factor out (-a) factor out (b) So

23. From Integers (Z) to Rationals (Q) • closure under addition/subtraction andclosure under multiplication/division • additive inverse and identity andmultiplicative inverse and identity • solution to all equations of the form for

24. a is a rational number if a = b/c for some integers b and c

25. Rational notations What is the same? What is different?

26. Rationals on the number line Take any two rational numbers (arbitrarily close) – can you find a third rational number that lies between them? Try this with fraction notation and with decimal notation.

27. Task: Terminating and recurring decimals When you convert fractions into decimals, some are terminating (with a finite number of decimal places) while others are recurring (with an infinitely repeated pattern of digits). • Which fractions make terminating and which make recurring decimals? Can you explain why? • Can you convert all terminating and recurring decimals into fractions? • What about non-terminating, non-recurring decimals?

28. “Doubling the square” • What is the length of the side of a square whose area is twice that of the unit square? Area = 2 square units Side length = ???

29. Suppose that √2 is rational. (p/q) = √2 where p and q are integers squaring both sides: (p/q)2 = 2 p2/q2=2 multiplying both sides by q2: p2=2q2 So 2 is a factor of p2 Each whole number has a unique factorisation into primes. Squaring a number doubles the number of occurrences of each factor, so in a prime factorisation of a square number each prime number occurs an even number of times. So 2 occurs at least once and an even number of times in the factorisation of p2, and 2 occurs an odd number of times in the prime factorisation of 2q2. If p2=2q2, then we have reached a contradiction. The argument is correct, so the assumption on which it was based must be false. Hence √2 cannot be written as a fraction p/q where p and q are integers. Hence √2 is irrational.

30. So √2 is irrational.What about … • √3, √4, √5, … ? • These are all solutions of equations of the formx2 = a (a is a positive rational number) • In general, solutions of polynomials with rational coefficients are algebraic numbers. Some irrational numbers are algebraic – including all surds – but others are not. • π is an example of a non-algebraic or transcendental irrational number. • Where are the irrational numbers on the number line?

31. Working with surds(irrational square roots) “simplest form” a + b√c where a and b are rational and c is the smallest whole number possible

32. How would you … • add and subtract • multiply • divide Giving your answers in simplest form.

33. “Rationalising the denominator” Find fractions equivalent to Find an equivalent fraction with a rational denominator. Find fractions equivalent to A rule for rationalising fractions is …

34. Find the sum What other similar sums lead to an exact whole number answer?

35. Resources • KS3 Number – Powers & Roots, Negative Numbers, More on numbers http://www.bbc.co.uk/schools/ks3bitesize/maths/number/index.shtml • KS4 – Number - Factors , powers and roots http://www.bbc.co.uk/schools/gcsebitesize/maths/number/ • KS5 – Core 1 and Additional Mathematics. Core 1 – Square roots and indices, Additional Mathematics – Expressions involving square roots some really good help on dealing with surds. http://www.meiresources.org/resources