© ATM 2013 • No reproduction (including Internet) except for legitimate academic purposes • [email protected] for permissions.
PERFECTHALVING,BICIMALS,AND
ATOAD-IN-THE-HOLE
JonnyGriffithsandMikeOllertonshowcasetheirtalentsbothculinaryandmathematical
T
hiswritingarosefromasequenceof
events,whichbeganatConference
2011conferenceinTelfordwitha
conversationabout‘bicimals’betweenJonny
andMike.ThisideadevelopedinOctober2011
whenMikecookeda toad in the hole for three
peopleandwhen,acoupleofdayslater,Jonny
sentMikesomeideashehadbeenworkingon
withhisYear12group.Thefollowingstoryisnot,
therefore,entirelyfictitious,thoughthetoad in
the holecouldjustaswellhavebeenmadewith
vegetarian,orevenCumberlandsausages…
Jonny:
Mike, that rectangular toad-in-the-hole you’ve
cooked looks fantastic. But, how are you going to
divide it up fairly? Andrew is coming around in a
minute, so there will be three of us. Now I know
that you are a perfect halver – you can halve
any portion perfectly and quickly. But how are
you going to use your halving skills to get three
perfect thirds?
Mike:
Well that’s easy Jonny. Being a perfect halver
means I’m also a perfect quarterer, eighther,
sixteenther, and so on. So I’ll make four perfect
quarters and we can have one piece each. Yes,
I know what you’re going to say next: what about
the remaining quarter? Well, I am just going
to repeat the same process again - see what
I mean?
Jonny:
Ah, neat! So, Mike, we each get a , then a
quarter of a quarter, which is a
, and then
th... and you are saying + + +... is a
third? Well, the first term is , and the number
we multiply by each time in the sequence, the
common ratio, is .
So, we have what I would call an infinite
geometric series, and one with first term a and
common ratio r has sum
as long as the size
of r is less than 1. In this case, the sum is
( )/(1 - ) =... ! It works, that’s brilliant!
However... I am sorry to put pressure on your
halving skills, Mike, but I’ve just had a text to say
two more friends are coming round, any second.
That’s five of us - how will your method produce
perfect fifths?
Mike:
Jonny you really must try to be less profligate
with your invitations. Just let me think... okay I’ve
got it. We’ll need to slice the toad into 8 equal
pieces so we all get one each. That leaves 3
pieces which I can halve into 6 sixteenths so
we can also have
each. This leaves a single
sixteenth size piece, so we are back to the
original problem of dividing one piece into five
equal portions - you do the maths, Jonny...
Jonny:
I just LOVE being told to do some maths! Okay,
Mike, so this time you are saying we each get:
+
+(
...) or +
x
+
)+(
+
x
+
)+(
+
x
...
x
Mmm... this looks like
( +
+
...) + ( +
+
...), that is two
infinite geometric series added together. So the
total sum will be
/(1 - ) + /(1 - ) = + = .
It works, so forgive me, Mike, for temporarily
doubting your halving abilities. But, can you get
any fraction by halving?
… And all these powers of two are making me
think in binary...
Mike:
Binary - now there’s a powerful system. A little
aside; on the 11th Nov 2011 or 11-11-11 I was
63 and it just so happens that 111111 in binary
is equal to 63 in denary - good eh? Anyway, I’m
also interested in ‘bicimals’ - they’re like decimals
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January 2013 www.atm.org.uk
29
© ATM 2013 • No reproduction (including Internet) except for legitimate academic purposes • [email protected] for permissions.
MOVINGBETWEENDIMENSIONS
only in binary. So after 16, 8, 4, 2, 1 we have a
bicimal point then we have , , , and so on.
Now then if we return to the time when it was
just you, me and Andrew who were going to eat
the toad, as a bicimal this would have come out
as 0.010101... or more succinctly 0.01 recurring.
So, in denary is 0.01 recurring in binary.
So what do you reckon would look like as a
bicimal?
Jonny:
Well, + +
+
+
+
... that gives
0.001100110011...? Another recurring bicimal,
but then every fraction has to be recurring... Let
me show you the same thing a different way.
There are five of us. So Mike, you repeatedly
halve the toad to give perfect sixteenths. We
take three each, which leaves one piece – where
you do the same again, and again... like you
did with making thirds. So we each have
an infinite geometric series with
, and
, so its sum is
) = . Now,
as a bicimal is
( ) / (1 0.0011, while ( ) is 0.00000011, and so they
add to give our recurring bicimal. Why does this
work? Because 5 divides exactly into 2 4 - 1, just
as 3 divides exactly into 2 2 - 1. So what happens
with 7? Now, before you get worried, Mike,
I haven’t invited anyone else round without
telling you...
Jonny:
0.000101110100010111010001011101...
Mike:
So that’s a recurring bicimal of period 10 - that’s
good. Furthermore, once we know what and
are as bicimals we can easily calculate and
by shifting all the numbers one place to the
left. So because = 0.01 then must be 0.10
recurring. Similarly because equals 0.0011
recurring then must be 0.0110 recurring. Also
to find we just need to add and and in
binary this is 0.1001 recurring.
Jonny:
Ah! I think that’s a knock at the door. Did I
mention that Andrew is a perfect trisector?
Mike:
Well that’ll just lead us into tricimals… so for a
start in base 3 is 0.1 recurring, perhaps we
should ask Andrew what he thinks would be…
Jonny Griffiths teaches at Paston College Norfolk
and Mike ollerton is a freelance consultant
Mike:
Phew, thank goodness for that. As for sevenths,
this is a piece of cake which, by the way, I
assume you are baking. Sevenths will be the
same process as thirds, except we just need to
slice the toad into eighths to begin with, then take
one piece each thus leaving one piece and then
we’re back to the original situation. In binary this
means is 0.001001001... or 0.001 all recurring.
Putting this information into the formula we have
/(1 - ), which clearly computes to .
Jonny:
...and this works because 7 divides into
2 3 - 1. Now elevenths...
Mike:
These are a real test of one’s patience...
Jonny:
Well, 11 divides into 2 10 - 1. So you halve the
toad into 1024 equal pieces, and the eleven
eaters take 93 pieces each, and we are left with
one piece that we can repeat this on.
Mike:
Now 93 in binary is 1011101, which means that
as a bicimal is...
30
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Mathematics Teaching 232 Journal of the Association of Teachers of Mathematics
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