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 (N)
• “counting numbers”
• 1, 2, 3, 4, …
Whole numbers = N + {0}
• 0    
what can you do within the set of
natural numbers?
• addition?
• subtraction?
• multiplication?
• division?
• solve the equations

x +5 1
x  3  10
Integers (Z)
• positive and negative and zero
• closure under addition and subtraction
• additive inverse and identity
a+0  a
a + (a)  0
• solution to all equations of the form
x + a  b (a,b  Z)
But …
• Negative numbers are difficult and have only
relatively recently been accepted in the West.
• see article by Jill Howard at
http://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?
-1+3=+2
+3
-4
-3
-3
-1-3=-4
-2
-1
0
+1
+2
+3
+4
signed numbers are
POSITIONS and MOVEMENTS
on the number line
-1+3=+2
+3
-4
-3
-3
-1-3=-4
-2
-1
0
+1
+2
+3
+4
signed numbers are
POSITIONS 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
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
-3
-2
-1
anything times zero …
x
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
-3
0
-2
0
-1
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 algebra
Define a number x as:
x  ab + a)b) + a)b)
Then
factor out (-a)
And
factor out (b)
x  ab + a)b) + b) x  a + a)b) + a)b)
x  ab + a)0)
x  0)b) + a)b)
x  ab + 0
x  0 + a)b)
x  ab
x  a)b)
So
ab  a)b)
From Integers (Z) to Rationals (Q)
• closure under addition/subtraction
and
closure under multiplication/division
• additive inverse and identity
and
multiplicative inverse and identity
• solution to all equations of the form
ax  b for a,b Q
a is a rational number
if
a = b/c
for some integers b and c
Rational notations
3
4
0.75
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?
1
1000
2
1000
Try this with fraction notation and with decimal
notation.


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 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.
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
1
2
Find an equivalent fraction with a rational
denominator.
1
Find fractions equivalent to
1 2
A rule for rationalising fractions is …
Find the sum
1
1
1
+
+L +
1+ 2
2+ 3
99 + 100
What other similar sums lead to an exact whole
number answer?
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