Structures 1

Number Systems

• What is a number?

• How might you describe the set of all numbers?

• How do you visualise the set of all numbers?

Natural numbers (

• “counting numbers”• 1, 2, 3, 4, …

Whole numbers = • …

what can you do within the set of natural numbers?

• addition?

• subtraction?

• multiplication?

• division?

• solve the equations

x 51

x 310

Integers ()

• positive and negative and zero• closure under addition and subtraction• additive inverse and identity

• solution to all equations of the form

x a b (a,b Z )

a 0a

a ( a)0

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

200 BC - Chinese number rods

132

5089

-704

-6027

• red – positive• black – negative

• Used in commercial and tax calculations

• black and red cancel each other out

• 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

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

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)

What difficulties do pupils have and how can you support them to make

meanings for negative numbers?

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?

0 +1 +2 +3 +4-1-2-3-4

+3

-3

-1+3=+2

-1-3=-4

signed numbers arePOSITIONS and MOVEMENTS

on the number line

signed numbers arePOSITIONS and MOVEMENTS

on the number line

0 +1 +2 +3 +4-1-2-3-4

+3

-3

-1+3=+2

-1-3=-4

signed numbers arePOSITIONS and MOVEMENTS

on the number line

signed numbers arePOSITIONS and MOVEMENTS

on the number line

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

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

Using a familiar pattern

x -3 -2 -1 0 +1 +2 +3 +4

+4 0 +4 +8 +12 +16

+3 0 +3 +6 +9 +12

+2 0 +2 +4 +6 +8

+1 0 +1 +2 +3 +4

0 0 0 0 0 0

-1

-2

-3

anything times zero …

x -3 -2 -1 0 +1 +2 +3 +4

+4 0 +4 +8 +12 +16

+3 0 +3 +6 +9 +12

+2 0 +2 +4 +6 +8

+1 0 +1 +2 +3 +4

0 0 0 0 0 0 0 0 0

-1 0

-2 0

-3 0

anything times one … (identity)

x -3 -2 -1 0 +1 +2 +3 +4

+4 0 +4 +8 +12 +16

+3 0 +3 +6 +9 +12

+2 0 +2 +4 +6 +8

+1 -3 -2 -1 0 +1 +2 +3 +4

0 0 0 0 0 0 0 0 0

-1 0 -1

-2 0 -2

-3 0 -3

complete the pattern …

x -3 -2 -1 0 +1 +2 +3 +4

+4 -12 -8 -4 0 +4 +8 +12 +16

+3 -9 -6 -3 0 +3 +6 +9 +12

+2 -6 -4 -2 0 +2 +4 +6 +8

+1 -3 -2 -1 0 +1 +2 +3 +4

0 0 0 0 0 0 0 0 0

-1 0 -1 -2 -3 -4

-2 0 -2 -4 -6 -8

-3 0 -3 -6 -9 -12

complete the pattern …

x -3 -2 -1 0 +1 +2 +3 +4

+4 -12 -8 -4 0 +4 +8 +12 +16

+3 -9 -6 -3 0 +3 +6 +9 +12

+2 -6 -4 -2 0 +2 +4 +6 +8

+1 -3 -2 -1 0 +1 +2 +3 +4

0 0 0 0 0 0 0 0 0

-1 +3 +2 +1 0 -1 -2 -3 -4

-2 +6 +4 +2 0 -2 -4 -6 -8

-3 +9 +6 +3 0 -3 -6 -9 -12

A proof using algebraDefine a number x as:

Then And

x ab a b a b

x ab a b b x ab a 0 x ab 0x ab

x a a b a b x 0 b a b x 0 a b x a b

So

ab a b

factor out (-a)factor out (-a) factor out (b)factor out (b)

From Integers (Z) to Rationals (Q)

• closure under addition/subtractionand

closure under multiplication/division• additive inverse and identity

andmultiplicative inverse and identity

• solution to all equations of the form

ax b for a,b Q

a is a rational numberif

a = b/cfor some integers b and c

Rational notations

3

40.75

What is the same?

What is different?

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.

11000

21000

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?

“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 = ???

Suppose that √2 is rational.(p/q) = √2 where p and q are integerssquaring both sides:(p/q)2 = 2p2/q2=2multiplying 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.

So √2 is irrational.What about …

• √3, √4, √5, … ?• These are all solutions of equations of the form

x2 = 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?

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

How would you …

• add and subtract

• multiply

• divide

Giving your answers in simplest form.

“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 …

1

2

1

1 2

Find the sum

What other similar sums lead to an exact whole number answer?

1

1 2

1

2 3L

1

99 100

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