What is Algebra? |

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Volume 1 Issue 1

One of the finest powers that a student can possess is the ability to evaluate a problem through reason. Reasoning is the capability of a student to determine the cause of an event from an effect observed. Another way of saying this is that reasoning is the ability of a person to draw conclusions from facts. When you reason you are doing work like a detective does in finding a murderer. A detective observes many different kinds of effects and facts in a murder case and from these facts he/she tries to determine who the murderer is. People in all walks of life use reasoning to do their work. A farmer must use reasoning to determine, before she plants seeds, whether or not she can profitably grow and sell her produce. In order to do this, the farmer must think a year or more ahead to determine whether or not she will make money on her crop.

The point is: Thinking, which involves reasoning (determining what is going to happen before it does), is what all of us do whether we realize it or not. Thinking, and then finding out that your were wrong, involves a lot of frustration, or work, in vain. This often leads to loss of money and time. For this reason, people want to follow sure, well know ways, to arrive at a practical answer, or solution to a problem that they are working on. If a way has been found by which something can be done better, or more quickly, then people want to know about it. People are always on the lookout for practical rules that can be used to help them solve their problems. In other words, people like to follow rules that other people discovered, that work.

When it comes to figuring things out that involve people it becomes very difficult to be correct in your reasoning all the time. This is because people can change their ways in the middle of a problem that you are trying to figure. One of the things that cannot be changed that way are problems involving numbers. Numbers cannot change when you solve a problem because a number is a

Algebra, then, is a system, or method, of determining the values of

As an example of a method of determining an unknown value from known values consider the following problem:

Determine the number that, when added to 14, equals 36?

The unknown number can be found by adding numbers to 14 until the number 36 results. Well, for example, 20 +14 = 34, which is 2 values below 36. So we determine,

22 + 14 = 36.

The problem may be solved by letting the unknown value (that we hope to determine) be represented by a letter of the alphabet, like a, b or c, or, maybe x, y or z. A number that is expressed by a

x + 14 = 36

Since this is an equality, that is, the left side of this equation equals the right side of the equation, the value of x may be determined by rearranging the equation in such a manner that the value of the unknown, x, is alone on one side of the equation and all the other values involved are on the other side of the equation. To get x alone on the left side of the equation we need to move the 14 from the left side of the equation. This can be done by subtracting 14 from both sides of the equation. For, if an equation is really a

x + 14 -14 = 36 -14

which can be written as

x = 36 - 14

or,

x = 22

So, through manipulation of the properties, or traits, of numbers (in this case using the arithmetic property, or trait, of subtraction), we have been able to determine an unknown value, x, from known values (36 and 14) in a simple equation.

Algebra uses letters of the alphabet, and sometimes, symbols such as Greek letters to represent numbers. But, why use a letter such as x to represent a number? What is the reason for using a letter or symbol to represent a number?

Consider the following to get an answer to these questions. Try to determine the significance of

x + y = z

Well, if we let x = 1 and y = 2, then, z = 3. Putting it another way,

1 + 2 = 3

Again, if we let x = 4 and y = 6, then, z = 10. This can be written as

4 + 6 = 10

Putting in other numbers for the values of x and y determines other values for z. If x = 220 and y = 47, then, z = 267. This can be written as

220 + 47 = 267

What are we getting at? It seems that all of the above additions

1 + 2 = 3

4 + 6 = 10

220 + 47 = 267

and, you can insert any other values for x and y and get their sum, z, are represented by the original equation

x + y = z

It appears that, if you look at x as

x + y = z

then, is a symbolism, or representation, or picture if you like, of any two numbers added to yield a third number. In other words, Instead of telling a person specific numbers that you want added, all you have to do is represent the above algebraic relationship and it shows the person exactly what you have in mind - without any numbers in it. It just shows them the process that you are referring to. That is the language of algebra. Algebra, then, can be looked upon as a generalization of a mathematical process you are referring, or directing someone's attention to. If you want to direct someone's attention to two numbers

x - y = z |

which expresses the concept that a number, y, subtracted from another number x equals a third number, z.

You don't have to use x, y and z to picture your problem. You could use a, b and c, as in

a - b = c

to represent the subtraction. Or, you could use

k - r = p

Or, you could use the Greek letters theta (2), chi (() and delta (*) to indicate, or picture, your subtraction, as in

2 - ( = *

This expression indicates that some number (, subtracted from another number 2 equals a third number (.

The concept of using algebraic letters or symbols to represent a mathematical process like addition, subtraction, multiplication, division, or any combination thereof, can be looked upon as a method by which one general algebraic formula can represent a mathematical process for finding a solution to a problem. In this method

As an example of an algebraic formula representing a combination of mathematical processes consider the relationship

4x + 6y(2r) = K

This relationship expresses the fact that "four times some quantity, x, + 6 times another quantity, y, times two times yet another quantity r equals a value represented by the letter K". Substituting values of x, y and r into the above equation will determine a value for the variable, K.

When a letter such as x appears in an algebraic expression the x is to be looked upon as a letter that is being used to represent, that is, to serve as a symbol or sign, that stands for, or, signifies, or, symbolizes a number that is under consideration. Care must be taken by the student to avoid confusing the concept of x as being something of importance all by itself. The letter x is a symbol that represents a number, just as a picture is a sheet of paper with an image on it that represents your friend. The picture is not the person shown on it. The picture is only a representation of a person. In the same way, x is not the number that you are dealing with at the time it is only a representation, or portrayal of the number. Because the a letter of the alphabet, such as x, can represents a concept, or consideration, of a phenomena, or object, whose value varies, like the gas in a motorboat varies, (becomes less and less) as the boat moves along the water, the letter is called a

Algebra can be used to represent all the mathematical operations like addition, subtraction, multiplication, division and combinations of these. On the previous pages of this newsletter we saw how algebra could be used to represent the mathematical properties of addition and subtraction. The number of terms in an expression could be increased and addition can be combined with subtraction as in

2x - 4z +3 y - 6m =142

This is a mathematical expression or statement of the fact that "2 times the value one number, x, minus 4 times the value of another number, z, plus 3 times yet another number y minus 6 times still another number, m, equal 142. The values of x, z, y and m must be known in order to properly execute this algebraic expression.

An

14x3 + 12y2 - 3z = t

the expressions 14x3, 12y2 and -3z are called

In algebra the mathematical operation of multiplication is expressed the same as multiplication is expressed in arithmetic. This is performed by positioning the quantities to be multiplied along side each other. If x and y are to be shown as multiplied together they would be shown as

xy

In order to avoid confusion the times symbol represented as an x is left out because of the common use of x as a function in algebra. Each of the quantities, or functions, involved in the multiplication shown above are called

Division of algebraic quantities is expressed the same as in arithmetic, that is, by placing the dividend (the quantity that is to be divided by another quantity) over the divisor (the quantity by which the dividend is to be divided), as

2 3 2

8x y /12xy z

If this expression is to be simplified the first thing to be done is to divide the numbers. In the fraction 8/12, the 8 can be divided by 4 and the 12 can be divided by 4 resulting in 8/12 = 2/3. Next, the x2/x factor can be divided to produce x2/x = x. Finally, the y3/y2 term can be divided to produce y3/y2 = y. The 1/z factor does not have a z in the numerator, and so, cannot be simplified any further. The substitution of the above divisions into the original equation produces

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