What is Algebra?
What is x?
X IS A SYMBOL FOR A NUMBER
How is Algebra Used to Express the Arithmetic
Student's Mathematical Review
Volume 1 Issue 1
What is Algebra?
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 constant, that is, a value that is invariable, or unchanging. Because numbers always have a specific value, rules can often be used to find answers, or solutions, locked within complicated combinations of numbers. This is a central reason why people became interested in algebra. Algebra was originally created as a system of reasoning for the purpose of determining, through calculation, what the value of certain unknown elements, or factors, are, when these elements are hidden within a combination of other factors and/or elements. Algebra is a branch of mathematics which presents methods by which a person can calculate unknown values of elements, that are hidden within a combination of known elements. Algebra uses a method of generalization of the number system, in which the properties, or traits, of numbers, and, the relationships between numbers, are expressed in terms of letters, symbols and numbers. The word algebra derives from the Arabic word al-jabr, specifically meaning the repair of broken parts, like in bone setting, but, referring to the concept, or inductive method of reasoning, of the reduction of parts to a whole.
Algebra, then, is a system, or method, of determining the values of unknown quantities from known quantities.
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 letter of the alphabet is sometimes called a literal. Let us call the unknown value x. We may represent the problem as
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 mathematical statement of equality. subtracting or adding the same value to both sides of the equation will not alter the equality present within the equation. Then,
x + 14 -14 = 36 -14
which can be written as
x = 36 - 14
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.
What is x?
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 any number and y as any other number added to the first number, then, z is a third number that is the sum of the two chosen numbers. The symbolism
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 subtracted you could use the relation
||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 any numbers can be substituted into the algebraic formula to arrive at a solution.
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.
X IS A SYMBOL FOR A NUMBER
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 variable. The main objective in algebra problems is to find the value of unknown variables. Often the statement used when a students is asked to solve a problem is stated as "Find the value of the unknown (variable)."
How is Algebra Used to Express the Arithmetic Operations?
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 algebraic expression is defined as a combination of ordinary numbers and letters (literal numbers) representing the basic operations of mathematics. Any algebraic expression picturing, or representing addition or subtraction is called an algebraic sum (subtraction is defined as the process of adding a negative number). The quantities between the plus and minus signs in an algebraic expression for a sum are called terms. ln an algebraic expression such as
14x3 + 12y2 - 3z = t
the expressions 14x3, 12y2 and -3z are called terms because they are separated from each by plus or minus signs. The 14x3 term is assumed to have a plus (+) sign before it, but, this is not shown with one as it is assumed to have one and it is not necessary to directly show this plus sign.
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
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 factors. In the above algebraic multiplication the quantity x is called a factor and the quantity y is also called a factor. The word factor should always be used when referring to an element or quantity in a multiplication operation. The quantities, or functions, present in a multiplication operation should not be called terms, because, in algebra, this word is reserved for quantities, or functions, that are added or subtracted.
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
Algebra is a collection of known methods of calculation by which unknown values can be determined.
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