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. © www.teachitmaths.co.uk 2014 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. 22231 © www.teachitmaths.co.uk 2014 ‘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… © www.teachitmaths.co.uk 2014 ! © www.teachitmaths.co.uk 2014 7 days in a week, 7 deadly sins, 7 dwarves, 7 wonders of the world. ∞ ! © www.teachitmaths.co.uk 2014 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. 22231 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 © www.teachitmaths.co.uk 2014 22231 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! 22231 © www.teachitmaths.co.uk 2014 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) = 22231 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. 22231 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 22231 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. © www.teachitmaths.co.uk 2014 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. 22231 © www.teachitmaths.co.uk 2014 42 is the answer to life, the universe and everything. 42 degrees is the angle at which light must reflect off water in order to create a rainbow. 22231 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. © www.teachitmaths.co.uk 2014 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. 22231 © www.teachitmaths.co.uk 2014 22231 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 traditional practice of extending yuletide celebrations over the twelve days from Christmas day to the eve of Epiphany. © www.teachitmaths.co.uk 2014 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. 22231 © www.teachitmaths.co.uk 2014 © www.teachitmaths.co.uk 2014 22231 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. 22231 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. Ask your maths teacher to check your answer! 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 © www.teachitmaths.co.uk 2014 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 What is i53 ? Ask your maths teacher! 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 22231 © www.teachitmaths.co.uk 2014 To multiply a 2 digit number by 11, add the digits and put the answer in between: 27 x 11 = 297 81 x 11 = 891 57 x 11 = 627 (carry the 1) 22231 There are 19 trailing zeros in 80 Roman XIX Binary 10011 Octal 23 Hexadecimal 13 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) © www.teachitmaths.co.uk 2014 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 22231 © www.teachitmaths.co.uk 2014 22231 3