MATHEMATICS ALL AROUND US Mathematics is a powerful tool that allows us to measure the world around us From simple systems of counting early mathematicians developed techniques such as arithmetic algebra and geometry to create calendars keep trade records or calculate taxes Today mathematicians use complex systems to dig into the deepest mysteries of the universe 4 1 2 10 9 5 6 3 4 7 8 Counting bases We usually count in a number system known as base 10 or the decimal system with one digit for each of our fingers and thumbs Sometimes we count in bases other than base 10 For example to count time we use base 60 Called the sexagesimal system it divides an hour into 60 minutes and a minute into 60 seconds This system was used by the ancient Babylonians 4,000 years ago Base 60 is useful because 60 is the smallest number that can be divided by 2 3 4 5 and 6 as well as 10 12 15 20 and 30 This makes it easy to divide an hour into equal parts
A clock face counts minutes in base 60 and hours in base 12 Solving equations An equation is a number sentence in which one side of the sign equals the other side 5 3 18 3 In algebra the unknown numbers in an equation are replaced by a letter The job of the mathematician is to solve the equation to find the value the letter represents y 7 2 5 therefore y 3 A formula is a set of symbols that expresses a mathematical rule Different values can be placed in the formula For example the formula to convert pounds into kilograms is y 5x 11 where y is the weight in kilograms and x is the weight in pounds The symbol means divided by 5 Mathematics in Wonderland By the 1800s new ideas had been introduced into mathematics that seemed impossible in the real world such as the square root of 1 English mathematician Lewis Carroll poked fun at many of these ideas in his 1865 novel Alice's Adventures in Wonderland In the book Alice is transported to a world in which impossible things happen all the time She tries to remember her times tables but finds that numbers behave very differently in Wonderland probably because she is no longer counting in base 10 Read on to discover the problems that mathematicians have solved through the ages The answers to each project's questions are found on page 31
RIGHT TRIANGLES All triangles have three interior angles that add up to 180 degrees In a right triangle one of those angles is always 90 Right triangles have many special properties that make them extremely useful to mathematicians The most important property was proven by the mathematician Pythagoras 6 Pythagoras about 570 500 bce The Greek philosopher Pythagoras ran a school in southern Italy He taught his students that the principles or rules of mathematics were the key to understanding the universe The stories we have about his life and teachings have been passed down in the form of legends and myths It is possible that either he or one of his students first wrote down the proof of his famous theorem The Pythagorean theorem The Pythagorean theorem states a2 b2 c2 Written as a sentence this means the square of the hypotenuse the side of a right triangle opposite the right angle shown at left as c is equal to the sum of the squares of the other two sides a and b In other words when squares are made on each of the three sides the biggest square has the exact same area as the other two squares put together c b a a2 b2 c2 b2 a2 c2
eye level height Distance from building Building height 24 13 25 12 5 4 3 5 7 40 41 15 17 8 9 Pythagorean triples Pythagorean triples are the lengths of the sides of right triangles with whole-number values Examples of Pythagorean triples are 3,4,5 5,12,13 7,24,25 8,15,17 9,40,41 Each of these triples has different interior angles Triangles that are different sizes but have the same angles are called similar triangles You can make similar triangles from the Pythagorean triples by multiplying each side by the same number For example if you multiply each side in the Pythagorean triple 3,4,5 by 2 you make a similar triangle 6,8,10 or by three you get another 9,12,15 Calculating heights Right triangles are used in the branch of mathematics called trigonometry Knowing the angles of a right triangle and the length of one of the sides allows you to calculate the lengths of the other sides Surveyors use trigonometry to calculate the heights of buildings among other things Using an instrument called a theodolite they measure the angle from a particular distance away from the building to the top of it If you move away from a building to a distance where the theodolite measures 45 you have created a right triangle with two sides of equal length You know the height of the building is equal to your distance from it plus the height of the theodolite above the ground 45º 45º 90º Distance from building 7 project VISUAL PROOF There are many ways to prove the Pythagorean theorem Pythagoras is said to have used this visual proof which you can recreate You will need a pen a card and a ruler 1 First cut out four identical right triangles 2 Arrange the triangles in a square with the four right angles at the four corners of the square Trace the square and cut it out 3 Inside the square the triangles form a smaller square with sides the length of the hypotenuse c 4 Rearrange the triangles so they form two rectangles within the same square The spaces left behind are two smaller squares with side lengths of a and b a a c b a b a b b a Do you see how this shows that a2 b2 c2
