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Number Systems
We are used to working with a decimal currency
system, based on ‘10’, but this was not always the
case.
In the UK we used to use ‘imperial currency’ (or
pounds, shillings and pence) in which a pound was
240 pence.
Many things were based on ‘12’ of something e.g.
‘a dozen eggs’, ‘12 inches in a foot’ and
‘12 pence in a shilling’.
Number Systems
Sometimes a different way of counting or
representing numbers can be useful. One of the
most common ‘alternative’ systems in use is
binary.
In the decimal system, we have ‘column headings’
which are powers of 10.
104
103
102 101 100
Number Systems
The binary system has column headings which are
powers of 2, as shown.
4
2
16
3
28
2
24
1
22
0
21
Number Systems
Can you work out what each of the following
numbers is?
24
23
22
21
20
Number
1
0
2
1
0
1
5
1
1
0
1
0
26
1
1
1
1
1
31
Number Systems
The binary system is the basis for a ‘magic’ trick
(cards on next slide) in which the audience
member has to choose a number and then say
which cards the number appears on.
The ‘magician’ then identifies the number.
• How does the magician do it?
• What numbers would go on the sixth card and
how would the others be amended?
A
1 3 5 7 9 11 13
15 17 19 21 23 25
27 29 31
B
2 3 6 7 10 11 14
15 18 19 22 23 26
27 30 31
C
4 5 6 7 12 13 14
15 20 21 22 23 28
29 30 31
D
8 9 10 11 12 13
14 15 24 25 26 27
28 29 30 31
E
16 17 18 19 20
21 22 23 24 25 26
27 28 29 30 31
F
Number tricks and proofs
There is a simple way to tell if a number is divisible
by 9.
Simply add the digits of the number together and if
the digit sum is divisible by 9, then the number
itself is divisible by 9.
19 is not divisible by
e.g.
9, so 1567 is also
1567 1+5+6+7 = 19
not divisible by 9
2367 2+3+6+7=18
19 is divisible by 9,
so 2367 is also
divisible by 9
Number tricks and proofs
Can you prove that this ‘rule’ is true for all two digit
numbers?
What about three digit numbers?
Number tricks and proofs
Hint
If we think of a two digit number such as 67, the 6
is in the ‘tens’ column, so a general two digit
number would be:
10
1
A
B
Number tricks and proofs
‘Indian’ two digit multiplication.
On the internet, a video shows an easy way to
multiple two digit numbers.
The example given is 18 x 14
The next slide shows how to do it.
Number tricks and proofs
18 x 14
Take the 4 from the 14 and add it to the 18
22 x 10 = 220
Now multiple the 8 and the 4
8 x 4 = 32
Add them together
220+32 = 252
Number tricks and proofs
Can you prove that this always works?
Number tricks and proofs
There are many ‘tricks’ to help multiply numbers
together. One method is to use your fingers to
help with the ‘trickier’ numbers such as 7 x 8.
It is interesting to try this out and then to see if you
can justify why it works.
• Number each finger and thumb as shown
8
9
10
7
7
6
8
9
6
10
• Put the 7 and the 8 together to form a ‘bridge’
9
10
8
7
6
6
• The ‘tens’ digit is given by the number of fingers
(and thumbs) making the bridge and under the
bridge.
In this case there are 5, so ‘50’
• The ‘ones’ digit is found by multiplying the
number of fingers above the bridge on one side
by the number above it on the other.
•
•
3x2=6
so with the ‘50’, that’s 56.
Teacher notes:
Number Systems and puzzles
In this edition we look at number systems and number ‘puzzles’.
The content is suitable for a wide range of ages and attainment levels
as the activities themselves are easily accessible, but the
corresponding reasoning and proof activity is often far more
challenging.
One of the activities also highlights the dangers of simply accepting a
method without really testing it or proving that it always works –
reasoning and justification are key!
Teacher notes: Number Systems
Binary card activity.
Simply add each of the powers of two (which is the first number of the
card) of all the cards the ‘secret’ number appears on.
Card F would have all of the numbers from 32 to 63.
Card A would have all the odd numbers.
Card B would have subsequent pairs of numbers
Card C would have subsequent groups of 4 numbers
etc.
Teacher notes: Number Systems
Multiples of 9
Writing a two digit number as ‘AB’ is really writing it as 10A+B
When the rule is used, one is checking whether A+B is a multiple of 9
or not. If it is then the whole number is a multiple of 9.
Starting with 10A+B, this can be re-written as 9A+(A+B)
9A is clearly a multiple of 9, so if A+B is a multiple of 9 then 10A+B will
also be a multiple of 9.
Similarly, with a three digit number ‘DEF’ this is really 100D+10E+F
This can be re-written as 99D+9E+(D+E+F)
Teacher notes: Number Systems
‘Indian’ two digit multiplication
The example given is 18 x 14.
The elements of this are 10x10, 8x10, 10x4, 8x4
Rearranging it to be (22x10) + (8x4) we essentially have
(10+8+4) x 10 + (8x4) which has the same elements.
However, what is not acknowledged in the video is that this only works
when the ‘tens’ digit is the same in both numbers. Breaking down the
elements should help to see why.
Exploring this should highlight to students the danger of simply picking
up something that ‘looks easy’ from the internet and using it.
Teacher notes: finger multiplication
Firstly, check what happens with 6 x 7 or with 6 x 6. There is a ‘tens’ to
carry from the unit multiplication.
It is a little challenging to prove this algebraically, so one way to prove it
is by exhaustion.
It can, however, be proved as follows:
• Consider the general case, a x b
a b
Teacher notes: finger multiplication
a b
For the ‘tens’ (on the bridge and below the
bridge)
On the left hand there are (a-5) fingers.
On the right hand there are (b-5) fingers
So a+b-10 will go in the ‘tens’ column.
Teacher notes: finger multiplication
a b
For the ‘units’ (above the bridge)
On the left hand there are (10-a) fingers.
On the right hand there are (10-b) fingers
(10-a)(10-b) will go in the ‘units’ column.
Teacher notes: finger multiplication
(10-a)(10-b) in the ‘units’ column.
a+b-10 in the ‘tens’ column, so this has to be multiplied by 10.
10(a+b-10) + (10-a)(10-b) =
10a+10b-100+100-10a-10b+ab=
ab
a b