# numbers of the week

```Number of the week
8 is a cube and the
only cube that is 1
less than a square.
26 = 43 = 82 = 64
The result when the
infinity symbol is
rotated 90°.
88888888 as a
string of digits occurs
46663520 places
after the decimal
point in π
8 is the base of the
octal system.
8 is the maximum
number of pieces a
cake can be cut into
using 3 slices.
8 perfect shuffles (‘riffle
shuffles’) will return a
deck of cards to its
original order.
64 is the number of
1 by 1 squares on a
chess board.
64 is the 32nd even
number, the 8th
square number and
the 4th cube
number.
8 times any
triangular
number is 1 less
than a square.
64 appears in π
as the 22nd and
23rd digits.
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‘When I’m 64’ is a
song from the Beatles
album Sgt Pepper’s
Lonely Hearts Club
Band.
e is a famous
irrational number,
and is one of the
most important
numbers in
mathematics.
7 is the only
dimension, besides 3,
in which a vector cross
product can be
defined.
Here are the first 32
digits:
2.7182818284590452
353602874713527…
!
7 days in a week,
7 dwarves,
7 wonders of the
world.
∞
!
7 is the start of an
arithmetic progression
of 6 primes:
7 37 67 97 127 157
7 colours are sufficient
to colour any map on a
torus.
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It is the solution to
the equation
=2
Manganese has
the atomic
number 25
In music, it is the
ratio of frequencies
of notes one
semitone apart.
It was first
discovered by Prince
of Zheng of the Ming
dynasty in 1584.
Orchestras tune
to A or 440 Hz.
A# has a
frequency of
440 x
1/12
1/12
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It is irrational just
like and e and so it
cannot be written as
a fraction.
1/12
2 x2 x2
x
1/12
1/12
1/12
2 x2 x2 x
1/12
1/12
1/12
x2 x2 x
2
1/12
1/12
1/12
x2 =
2 x2
2
December 25th
is Christmas
day!
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A regular
heptagon
is the smallest
polygon that can’t
be constructed
using a ruler and
a compass.
25 = 4! + 1
There are 25
prophets
mentioned in
the Quran.
25
is the smallest square
number that is the sum
of 2 non-zero squares
25² = 3² + 4²
0. 142857
P(7) =
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Fermat asserted
correctly that
25 is the only square
that is 2 less
than a cube (3³ - 2)
All powers of 25
end in the digits 25
!
7 is the smallest prime
whose reciprocal is of
maximum length
When rolling two
standard 6-sided
dice, 7 is the most
likely total.
It is often known as
Euler's number after
Leonhard Euler
(1707-83).
1
⋯,
∞
!
!
!
therefore e can be expressed as:
⋯
e 1
64 is written as
1000000 in binary.
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f(x) = ex
f’(x) = ex
f”(x) = ex
e is the base of the
natural logarithms;
invented by John
Napier in 1614.
10 is the base for
common logarithms.
64 is the smallest
number with 7
divisors.
Also known as
the ‘twelfth root
of 2’ or
As a decimal it is
1.059463094359…
It can also be
expressed as
2
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1
Number of the week
In any Fibonacci sequence,
successive terms get closer to
the golden ratio:
1 1 2 3 5 8 13 21 34 55
3÷2=1.5
5÷3=1.66666…
8÷5=1.6
13÷8=1.625
21÷13 = 1.61538…
34÷21=1.61904…
This number is
called the
‘golden ratio’ or
the ‘divine
proportion’.
As a decimal it
can be written as
1.618033989…
A
If
then
The Starship
Enterprise has 42
decks.
B
C
will equal the golden ratio
The golden ratio is used in architecture
as it is supposedly the ratio of length to
width of an aesthetically pleasing
rectangle.
There are exactly 42
dots on a pair of
dice.
42 is a PRONIC
number (product of
2 consecutive
numbers).
The golden ratio appears
all around you. Here are
some examples:
• Leonardo Da Vinci’s Mona
Lisa
• the Parthenon in Greece
• length between finger tip
and elbow to wrist and
elbow.
• the ratio of lengths in a
pentagram.
The right arm of
the Statue of
Liberty is 42 feet
long.
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life, the universe and
everything.
42 degrees is the angle at which light must
reflect off water in order to create a
rainbow.
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The number of sides
in a tricosagon.
With 1018, it is one of only two powers of 10
between 109 and 105000 that can be written
as the product of two zero free factors.
23! is 23 digits long.
If there are 23 or
more people in a
room, then the
probability of a
shared birthday is
more than 0.5
The number of
goals Ronaldinho
once scored in a
23-0 victory.
8 589 934 592
116 415 321 829 934 814 453 125
For every n greater
than 23 none of the
binomial coefficients
nC are square free.
r
This is a six digit number that is equal
to the sum of the sixth powers of each
of its digits.
548 834 = 56 + 46 + 86 + 86 + 36 + 46
23 is the 4th prime whose reciprocal is of
maximum length
1/23 = 0.0434782608695652173913
There are other six digit numbers with
this property – can you find them? You
may need to use a computer.
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The atomic number
of magnesium in the
periodic table.
Twelve is a
composite number,
the smallest number
with exactly six
divisors,
its divisors being
1, 2, 3, 4, 6 and 12
A twelve-sided
polygon is a
dodecagon. A twelvefaced polyhedron is a
dodecahedron.
Regular cubes and
octahedrons both
have 12 edges.
There are 12 days of Christmas. The song
‘Twelve Days of Christmas’ came from the
celebrations over the twelve days from
Christmas day to the eve of Epiphany.
10 is 1010 in binary.
Number of dots in a
tetractys.
A polygon with 10
sides is a decagon,
and 10 is a
decagonal number.
The duodecimal
system is the use of
12 as a division
factor. It was used
for many medieval
weights and
measures, including
hours.
10 is the sum of
the first three
prime numbers.
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10 is the number of
hydrogen atoms in
butane, a
hydrocarbon.
In association football, the number 10 is
traditionally worn by the team's playmaker.
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2
Number of the week
The equation
x2 + 4 = 0
has solutions 2i and
-2i
The 18th century
Swiss mathematician
Leonhard Euler
boldly proved that
1
(1 + 3i)(2 - 7i)
can be multiplied
out using FOIL.
teacher to check
It can be shown
that
ii =
which as a
decimal is
0.2078795763…
Apollo 11 was
the first
manned mission
to the moon’s
surface.
The imaginary
number can be
used to prove many
important results like:
cos(5x) = 16(cosx)5 20(cosx)3 + 5cosx
The escape
velocity of the
Earth is 11
km/s.
11 is a prime
number. It has
2 factors.
i=
1
i2 = -1
3
i =-i
i4 = 1
11 is the largest multiplicative persistence.
Multiply the digits of any number
together and do the same with the
answer until you end up with a single digit.
You won’t be able to do this any more than 11
times. It is conjectured that the largest
number (without 1’s) that has this property is
777777333322222222222222222222
i is called the
‘imaginary number’
i=
1
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To multiply a 2
digit number by
digits and put
between:
27 x 11 = 297
81 x 11 = 891
57 x 11 = 627
(carry the 1)
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There are 19
trailing zeros in 80
Roman XIX
Binary 10011
Octal 23
19 is a ‘happy’
number
1²+9²=82
8²+2²=68
6²+8²=100
1²+0²+0²=1
19 the smallest
prime that equals the
(product of the digits)
+ (sum of digits)
It is the smallest
prime number that
is the sum of three
primes
19 = 3 + 5 + 11
Pascal found 19 cases of number
patterns in his famous triangle.
Pascal was born on 19-6-1623
and died on 19-8-1662
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