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What is Algebra

A branch of mathematics in which arithmatical operations are generaized by using alphabetic symbols to represent unkown numbers.

That's the dictionary definition, but what does it mean. Well algebra is a mathematicians shorthand, lets take an example.

You buy 10 apples in the supermarket for 2.00, how much does one cost. That's very simple but how would you write that down on paper?

Like this maybe: 10 * apple = 2.00 (* meaning times)

But a mathematician might write it like this: 10a = 2.00

The mathematician is very lazy, he does not want to write the whole word "apple" so he decides just to write the single letter a as a symbol for an apple, also he does not want to write the multiply (times) sign as he knows 10a means 10 times a.

Buy the way "10a = 2.00" is said to be an equation, an equation always contains one and only one equals sign (=). The equals sign means whatever is to the left of the equal sign is the same as everything to the right of the equal sign. In this case we are saying that the value (cost) of the 10 apples is the same as 2.00.

If in addition you bought 5 bananas then the mathematician would write this as 5b, but what if as well as the apples and bananas you bought two avocados. If you had not bought any apples then the mathematician would use a as a symbol for an avocado, but as he has already used a as a symbol for an apple so using a as a symbol for an avocados would be confusing. For example he might write down an a peace of paper:

10a + 5b + 2a = 4.00

Then tomorrow he has forgotten what was bought and looks at the peace of paper he wrote what was bought and scratches his head, you definitely bought 5 bananas but did you buy 10 apples and two avocados or 10 avocados and 2 apples. This illustrates a very important point in algebra, every symbol you use must have only one meaning, you must not use the same symbol to represent (mean) two or more different things. In this case when deciding on the symbol to use for an avocado the mathematician might say to himself "I have used the symbols a and b so I will use the symbol c for an avocado".

You are not stuck now with always using the symbol c to stand for an avocado, it is OK to use the symbol a for an avocado tomorrow if you do not buy any apples. Well hopefully it is OK but if at the end of the week you are going to take all the pieces of paper the mathematician wrote down what was bought on, to see what you bought in total for the week. You would be in trouble if on one day you used the symbol a to stand for an apple and on another day you used the symbol a to stand for an avocado. In this case you need to be careful and use the same symbols each day so that at the end of the week you can easily look at all the equations for each day and determine the total number of apples and avocados you bought.

So a mathematician creates algebra equations as a shorthand for a balance of scalars in the real world. A scalar, what is that, well it is a quantity of something. In the first equation we wrote down above "10a = 2.00" the a is not a symbol for the object an apple but for the scalar the price of the apple and the equation says 10 times the cost of one apple is 2.00. To solve this equation we know that if 10 apples cost 2.00 one apple costs 2.00 divided by 10, that is 20 pence. In algebra terms what are we doing, on the left side of the equation we have 10a, we want to know what 1a is so we need one tenth of the left side of the equation. The equals sign means that everything on the left of equal sign is the same as everything on the right, we want to know what one tenth of the left side of equation so if we take one tenth of right side of equation the equation will remain in balance. Taking one tenth of both sides of equation means we get a new equation that looks like this a = 2.00/10 or in other words a = 0.20.

We have just done a little bit of algebra, that is we have an algebra equation and from this equation we have created a new equation by dividing both sides of equation by 10. In doing this we have discovered what the value a, which is a symbol representing the cost of an apple, is and this is said to be solving the equation.

Algebra then is the application of arithmetic operations to an algebraic equation with the aim to solve or simplify it.

Please note that these arithmetic operations may not just involve numbers but can involve algebraic symbols. In our equation above instead of dividing by 10 we could have subtracted 9a from both sides of the equation, as we are subtracting the same amount from both sides of the equation the equation will remain balanced and we will get a new equation 10a - 9a = 2.00 - 9a. Its simple to see that 10a - 9a is 1a so the equation becomes a = 2.00 - 9a unhappily this does not help us discover what the value of a is, sometimes an operation like this does help. The hyperlink What Is Simplification below goes much further into algebraic arithmetic.

What is Simplification

Algebra Cheat Homepage

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