### Number Facts

In addition to providing some useful hints, our flash cards also contain some interesting facts about the numbers involved in the question. There is only so much information that can be fitted onto the back of a card, so in case you've spotted a fact and would like a bit more information, we've added some details below with some links to other resources where you can find out even more info. Beware, once you start looking up number facts it can get quite addictive.

#### Ancient Egypt

The ancient Egyptians were talented mathematicians, even though the number system they used didn't make manipulation of numbers very easy. It is obvious from the Egyptian structures that still exist such as the great pyramids, that they must have been good at maths. In addition to these fantastic structures, a number of documents have been found that show how important teaching maths was to Egyptian society. For example the Rhind Mathematical Papyrus, which can be seen in the British Museum contains 84 problems, including problems that you are likely to see in a class today.

Although the Egyptians used a decimal system in the same way that we do today, they only had seven symbols which were as follows:

- 1 is a single line, much like the 1 we use, but there wasn't a separate symbol for 2,3,4,5,6,7,8 or 9 so to represent these they would write the correct number of lines
- 10 was represented by an n shape that represented the a heel bone
- 100 was represented by a coil of rope
- 1,000 was a drawing of a lotus plant
- 10,000 was a drawing of a finger
- 100,000 was a drawing of a tadpole / frog
- 100,000 was the figure of a god with arms raised above his head

So writing big numbers became quite a task. For example to write 4622 in Egyptian hieroglyphs you would need to draw the following:

Four lotus plants (4000), six coils of rope (600), 2 heel bones (20) and finally two lines.

Although this would look nicer than our numbers, it would take ages to write out, especially if you had to write it onto papyrus.

If you'd like to know more about Egyptian mathematics and perhaps tackle some problems using Egyptian numbers, why not try this site: www.eyelid.co.uk/numbers.htm

#### Colouring a Map

In mathematics, the four colour theorem, or the four colour map theorem, states that given any separation of a plane into contiguous regions, called a map, the regions can be coloured using at most four colours so that no two adjacent regions have the same colour. Two regions are called adjacent only if they share a border segment, not just a point.

This was first suggested in 1852 by South African mathematician Francis Guthrie while trying to colour the map of counties in England. It took 124 years and the invention of the computer before this was finally proved to be true in 1976 by Kenneth Appel and Wolfgang Haken. To this date there are still some that have doubts about this proof, so if you draw out a map and find that you need more than four colours to colour it in so that no two adjacent regions are the same colour, then please let us know.

#### "Forty" in English

The letters of the English word "forty" are in alphabetical order. This is the only number in the English language where this is true. Why not see if you can prove that this is true (it's OK, you wont need to write out every number there is).

#### Hexadecimal

Hexadecimal or hex is the base 16 number system. What this means is that rather than the 10 symbols that we generally use in the standard decimal system the hexadecimal system uses 16 different characters. The first ten symbols are the same as decimal i.e. 0-9, after that an A is generally used to represent 10, B=11, C=12, D=13, E=14 and F=15. So rather than units, tens and hundreds columns that we get with a decimal system in a hexadecimal system we have a units column, a 16s column and a 256s column (16*16) etc etc. This means that if you wanted to write 33 in hexadecimal you would actually write 21, which is 2*16 plus 1 unit (32+1=33).

This is all very interesting, but why would you use it. Well all computers work in base 2 (binary) which is lots of 1s and 0s. Humans find binary very difficult to deal with, since binary numbers get very big very quickly. For example, 33 in binary is actually written as 100001. Since it is much easier to map binary numbers to hexadecimal numbers than it is to decimal numbers, hex is used a lot in computer science. One example is colours on web pages which are often described using hex number such as #CDB7B5. These numbers can be split into three pairs where the first two characters describe how much red should be used, then second how much green and the third how much blue there should be. Using this method it's possible to describe 16.7 million colours with only 6 characters.

#### Hitchhiker's Guide to the Galaxy

Originally a BBC Radio 4 comedy first broadcast in 1978 and then went onto become a series of very funny science fiction books. In 2004 the film of the first book was finally released. Through a series of strange events the main characters in the story eventually find a super computer that has been trying to work out the answer to the ultimate question to life, the universe and everything. After pondering this question for seven and a half million years, it declared that the answer was **42**!!

I could go on, but if you want to know more then pick up a copy of the book or even better get a copy of the radio series. You wont regret it.

#### Huli People - Base 15

Huli is a Papuan language spoken by the Huli people of the Southern Highlands province of Papua New Guinea. The language features a quindecimal (base-15) numeral system, which means that rather than counting in tens they count in 15s. It isn't known exactly why the Huli speakers adopted a base-15 system, but it is thought that in addition to counting on their hands they also included other body parts in addition to use of the hands, such as head and chest to get up to 15. See this paper for more information Counting and Number in Huli.

#### Human Hand Bones

The human hand has **27** bones. There are 8 in the wrist, the palm contains 5 and there are three bones in each finger and two in the thumb. In a typical adult human there are 206 bones, so the two hands account for just over a quarter of all the bones in the body.

#### Mayan and Aztec - Base 20

The ancient Mayans and Aztecs used a base 20 number system. It is thought that most ancient people who adopted a base 20 system did so because they counted on their fingers and toes. Although they used a base 20 system, they only used three different symbols unlike the 10 different numbers we use to count with. They used a line (probably representing a stick) for 5, a dot (a pebble) for a unit and a shell shape for zero. So for 8 it is three dots above a line.

#### Mesopotamians - Base 60

The base 60 numeral system was first used by the ancient Sumerians in Mesopotamia over 5000 years ago. It isn't known why they used a base 60 number system, but one thought is that it was chosen because 60 has so many divisors (1,2,3,4,5,6,10,12,15,20,30,60) and is the smallest number that has 1,2,3,4,5 and 6 as divisors. This meant that it was very easy to manipulate fractions in base 60.

Later as the Babylonians took control of Mesopotamia, they also adopted this base 60 system but they also crucially introduced the idea of a place system, so the value of the number depended on where they are placed. This is similar to what we use today. When we write a 1 and then move it one place to the left it then becomes 10. This was a revolution that allowed the Babylonians to tackle complex mathematical problems. These advances in mathematics greatly influenced future mathematicians from many cultures and is the reason why we still use a base 60 system for time (60 minutes in and hour and 60 seconds in a minute) and measuring angles (360 degrees in a circle, which is 6 * 60). This modern usage of base 60 can also be seen in some modern languages. For example in French the number 70 is soixante-dix, where soixante means 60 and dix means 10, so 70 is actually 60+10.

#### Moon Orbit

The moon takes approximately 28 days to orbit the earth. Because it also spins on its axis as it orbits the earth, we only ever see one side of the moon from the earth.

#### Perfect Numbers

A Perfect Number is a whole number whose factors (not including itself) add up to that number. The first perfect umber is **6**, since its factors when added together equal 6 (1+2+3=6). The next perfect number is 28. See if you can work out whether this is true or not. What are its factors?

People have been studying perfect numbers for thousands of years. The ancient Greek mathematicians only knew about the first four perfect numbers (6, 28, 496 and 8128), which isn't too bad when you realise that the next one is 33,550,336. To date we have discovered, thanks to computers, about 44 perfect numbers the largest of which has almost 20 million digits, so i wont write it out here.

#### Prime Numbers

A Prime Number is any number that has just two factors. 1 and itself. The first 25 prime numbers are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

In maths knowing which numbers are prime can be very useful, since you know when you get a prime that it only has two factors, 1 and itself so you can't divide it by anything else.

Without prime numbers we wouldn't be able to securely shop on the internet. Large prime numbers form the basis of security on the internet and are used to encrypt important information such as credit card numbers as they are sent across the web. The process is quite simple, but since it is very difficult to work out what the factors are of a large number, this is what makes it secure. It is not impossible to break this sort of security, but it would take the best computers in the world a long time to crack the encryption, which makes it impractical to try and break the code.

#### Roman Numerals

The Romans used a decimal system like the one that is commonly used today, but unlike us and the ancient Babylonians, they didn't use a place value system, which meant calculating sums with Roman numerals was difficult. The Roman numeral system consisten of seven symbols:

Symbol | Value |
---|---|

I | 1 |

V | 5 |

X | 10 |

L | 50 |

C | 100 |

D | 500 |

M | 1000 |

#### Square Numbers

In mathematics a square number is a positive whole number that is the result of multiplying another whole number by itself. Fir example 9 is a square number since 3 * 3 = 9. The first eleven square numbers are:

- 0
^{2}= 0 - 1
^{2}= 1 - 2
^{2}= 4 - 3
^{2}= 9 - 4
^{2}= 16 - 5
^{2}= 25 - 6
^{2}= 36 - 7
^{2}= 49 - 8
^{2}= 64 - 9
^{2}= 81 - 10
^{2}= 100

#### Teeth

The average human has 32 teeth, which is the same as a Giraffe but a lot less than a shark which lose teeth each week and may have over 20,000 teeth in a lifetime.

#### Tennis Scoring

In tennis a match is one when a player wins the best of 3 or 5 sets. Each set is made up of a number of games and each game has a rather strange point system. The first four points are scored as love (zero), 15, 30 and 40. The next point wins the game.

The origins of this scoring system aren't known, but it's possible that it relates to the medieval love of the number 60. Where 15, is the first quarter, 30 is the second, 45 (abbreviated to 40) is the third and then the final quarter is the end of the game.

#### Tricontagon

A polygon is a two dimensional (plane) shape with a number of straight sides. The word "polygon" derives from the Greek πολύς ("many") and γωνία (gōnia), meaning "knee" or "angle", although now we tend to think of it as "many sides" rather than "many angles" and definitely not "many knees".

A triangle is an example of a polygon with 3 straight sides. A tricontagon is a polygon with 30 sides, since "triconta" means 30. Below is a table of some other names for polygons, some of which you will probably recognise:

Name | Number of Sides | Comments |
---|---|---|

Triangle | 3 | |

Quadrilateral | 4 | A square and a rectangle are examples of quadrilaterals. |

Pentagon | 5 | In the United States the Department of Defence are located in a 5 sided building in Virginia called "The Pentagon". In sport the modern Pentathlon is a sport that includes the shooting, swimming, fencing, show jumping and cross country running. So five sports in all hence the "pent" in pentathlon. |

Hexagon | 6 | |

Heptagon | 7 | In the UK both the 50p coin and the 20p coin are heptagon shape. In sport the womens outdoor heptathlon consists of the following 7 events: 100m hurdles, high jump, shot put, 200m race, long jump, javelin and 800m race. So seven sports in all, hence the "hept" in heptathlon. |

Octagon | 8 | An Octogon has eight sides and an Octopus has eight legs. Can you sport the similarity in these words? |

Enneagon | 9 | |

Decagon | 10 | In olympic sport there is an event tat consists of the following 10 events: 100m race, long jump, shot put, high jump, 400m race, 110m hurdles, discus, pole vault, javelin and 1500m race. Apart from exhausting, can you work out what this event is called? Yep a decathlon. |

Hendecagon | 11 | |

Dodecagon | 12 | |

Tridecagon | 13 | |

Tetradecagon | 14 | |

Pentadecagon | 15 | |

Hexadecagon | 16 | |

Heptadecagon | 17 | |

Octadecagon | 18 | |

Enneadecagon | 19 | |

Icosagon | 20 |

#### Yuki Pomo - Base 8

A base-eight system was devised by (at least) the Yuki Pomo of Northern California, who used the spaces between the fingers to count. I guess that if the Simpsons decided to invent a numeric system they would also choose base 8. Do you know why?