1 natural number A the proof that something is mathematically true
2 integer B a number larger than zero
3 operation C any number (positive or negative) or zero
4 function D a way numbers combine together
5 right angle E the relationship between argument and result in calculus
6 square F the result of combining numbers
7 sum G an angle of 90 degrees
8 satellite navigation system H system in orbit around the Earth for directions
9 theorem I the product of two equal terms
10axiom J principle
What mathematicians study can be summed up as relating to four major fields. They look at quantities - how much or how many. There is also the study of structure - how things are arranged together and the relationship between the parts. Then there is the study of space, where mathematicians are interested in the shape of things. Finally, there is change and how things move, over time or through space.
Quantity is mostly concerned with numbers. Mathematicians are interested in both natural numbers and integers. Natural numbers are those which are greater than zero, while integers may be zero itself or more or less than zero. There are four ways these may combine together; these are called operations. In arithmetic, we know the operations as addition (+), subtraction (-), division (:) and multiplication (x). For a fuller, more philosophical understanding of number and the operations that can be applied to them, mathematicians look to Number Theory.
The study of the structure of things is said to have begun with the Greek mathematician, Pythagoras, who lived from 582 to 507 BC. Every mathematician has to learn his most famous theorem. A theorem is a proof of mathematical truth. Pythagoras showed us that in a right-angled triangle, the square of the side of the hypotenuse is equal to the sum of the squares of the other two sides. The hypotenuse is the longest side of such a triangle and that length, multiplied by itself is the same as the length of one side multiplied by itself and added to the other side multiplied by itself. Mathematicians find it easier to write this as: a2+b2=c2, (a squared plus b squared equals c squared) where c is the hypotenuse. It is Pythagoras' Theorem which gives us algebra, a branch of mathematics that originated in the Arab world.
Another Greek mathematician laid the foundations for our understanding of space. More than 200 years after Pythagoras, Euclid used a small set of axioms to prove more theorems. This, we know today as geometry. He saw the world in three dimensions - height, width and length. Developments in other sciences, most notably in physics, have led mathematicians to add to Euclid's work. Since Einstein, mathematicians have added a fourth dimension, time, to Euclid's three. By combining space with number we have developed the trigonometry used in making maps both on paper and in satellite navigation systems.
From algebra and geometry comes calculus. This is the most important tool that mathematicians have to describe change, for example, if you calculate the speed of a moving car or analyse the way the population of a city changes over time. The most significant area of calculus is function, which is concerned with the relationship between argument and result. Indeed, the field of functional analysis has its most important application in quantum mechanics, which gives us the basis for our study of physics and chemistry today.
There is more to maths than this. For example, pure maths involves a more creative approach to the science. An important field of study is statistics which uses Probability Theory, the mathematical study of chance, to predict results and analyse information. Many statisticians would say that they are not mathematicians, but analysts. However, without maths, statisticians would all agree, there would be no statistics at all.