Algebraic Notation

The most obvious feature of algebra is the use of special notation.  Symbols, called variables,  are used to  to represent numbers. Variables are usually letters such as x, y, z or a and b. Special symbols such as "=" and " " are used to denote relationships (equality and less-than-or equal). Other symbols are used to denote the familiar arithmetic operations: + for addition;  or juxtaposition for multiplication ;  -  for subtraction;  or / for division; and for positive square root.  Superscripts (exponents), such as the 2 in x2, are used to denote repeated multiplications. For example, x2 is a shorthand for xx, and x3 is a shorthand for x.x ).  The symbol  (the Greek letter called "pi")  is used to represent the special number  3.14159...

Thus algebra is full of expressions such as

3x2 + 2x + 4
(a+b)/2
a +c
and  equations (pairs of expressions separated by an equality sign, "=")
y = 2x3 + 2x + 7
mn = nm
(x+a)(x-a) = x2 - a2
A = r2

What are variables?

The letters that occur in algebraic expressions, such as  the x in 3x2 + 2x +4,  are called variables.   The numbers, such as the 3, 2 and 4, in the above expression are constants  -- their value is constant..  But the variables "vary", that is, they can be replaced any number to get an arithmetic expression (an expression just involving numbers and operations)which can then be evaluated.
If in the expression 3x2 + 2x +4  we take x = and  then we get the result
To see the value of the expression for other values of x just change the value of x  and  click on the "evaluate the expression" button.

Note: if the value you give to x is too large then you won't be able to see all of the answer unless you use the arrow keys.

Note: For some calculations involving decimals the answer may be "slightly off" -- this is a result of the way in which the computer handles decimals.  For a simple example of this take x = 5.3.

What use are algebraic expressions and equations?

One use of such expressions is to provide a succinct way to express certain problems,  formulas and general mathematical facts.  Consider the following three statements:

1. find the number such that 3 plus that number equals 7
2. the area of a circle is given by the product of the number  (approx.  3.14159)   with the product of the radius of the circle with itself
3. the product of two fractions is the fraction whose denominator is the product of the denominators of the two original fractions and whose numerator is the product of the numerators of the two original fractions.
4. given any two numbers we get the same result regardless of which order in which we add them .
Using algebraic  notation we can rewrite these as, respectively
1. find x such that  3+x = 7 .
2. A = r2
3. n/m p/q  =  np/mq
4. for all numbers n and m, n+m = m+n
Equally important, this notation makes it easy to manipulate the statements in a manner that leads to solutions of problems and/or to new results.

The formula A = r2 expresses the area, A,  of a circle in terms of its radius r.  That is, given the radius of a circle we can use the formula to find its area.  For example, a circle with a three foot radius  has an area  A = 32 33 = 9,  so, taking 3.14 as our approximation for ,  we get  A = 3.149 = 28.26 as the area of the circle. What we have done is replace the "variable" r with the specific number, 3, and the constant symbol, , with its (approximate) numerical value, and evaluated the expression to get a corresponding numerical value for the variable A, that is, the area of the circle..

But what is our problem is that we know the area and we want to find the radius?  For example, if a gardener has enough seed to plant 50 square feet of some flower and he wants to plant it in a circular bed then how big a flower bed should he dig?  In this case we can use the same formula, A = r2, but this time we know the area, A, and we are trying to find the radius, r.   That is, we know that A = 50 = r2.   Again, approximating  by 3.14, we can rewrite this as 50 = 3.14r2.  Dividing both sides by 3.14 we get 15.9236 = r2,  then, taking the square root of both sides gives 3.99 = r. Thus, from the practical point of view (of the gardener) the desired flower bed should have a radius of four feet!

We can get a general formula for the desired result from the formula A = r2 by dividing both sides of the equation by  , getting A/ = r2 and then taking the square root of both sides getting the general formula r =A/, for the radius when given the area.. Note that this formula says to do just what we did before.  That is,  solve the gardeners problem we take A = 50, divide it by  and take the square root.    Of course you may be wondering, just how did we know what to do?  Well, that is one thing you will learn by studying  High School Algebra.

Modern algebraic notation offers many advantages over earlier notations.   Primary among these is that it makes it easier to manipulate algebraic expressions  and thus makes it easier to explain and  apply algebraic concepts.

History of Algebraic Notation

The history of algebraic notation  is long and varied going back over thousands of years.  Our present notation is a relatively recent development going back only several hundred years.

The use of symbols, equality  signs and superscripts is actually a relatively recent development in  algebra .  Through most of its history algebra was written out using just words and numbers.

The "modern notation" makes it much easier to write down algebraic ideas and results.  More important,  the use of appropriate notation makes it much easier to perform many of the routine manipulations needed to solve algebraic problems.

Without the use of modern notation it would probably be extremely difficult to write a computer program for the polynomial calculator.

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Algebraic notation has a variety of uses.  For example, we can use it to write formulas such as the formula A = r2 for the area of a circle,  to give mathematical definitions such as (n/p + r/s) = (ns + pr)/sr for the addition of fractions ,  to present general mathematical results such as m+n = n+m  expressing the commutativity of  addition (the fact that the result of adding two numbers does not depend on the order in which we add them),  and to describe many kinds of functions.  Thus algebraic notation provides a concise way to express many kinds of relationships.

However, even more important is the fact that algorithm notation can be used to manipulate such expressions so that we can solve a variety of problems.  Much of the material in high school algebra is concerned with various ways to manipulate such expressions and with the different kinds of problems we can solve with such manipulations.

Is mathematical notation perfected?

Since mathematical notation has been being improved for several thousand years we might expect it to be pretty well perfect by this time.  Unfortunately this is not really the case and there are many situations where the notation is potentially ambiguous.   That is, there are many situations in mathematics where a given notation can have more than one meaning.  Often the different meanings are closely related but the differences are still significant.   This ambiguity does not cause problems for experienced mathematicians since they can generally figure out what is meant from the overall context, but it can cause some real problems for students.

• What is the meaning of  2/4?
• What is the meaning of   "="