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May 25th

May 25th

# Literal Numbers

The basic difference between algebra and simple numerical arithmetic is that in algebra we can use written or literal symbols to represent numerical values. These symbols are often ordinary alphabetic characters: a, b, c, d, … or A, B, C, D, …. . (Usually in algebra, uppercase characters are not considered to be the same as lowercase characters – ‘A’ and ‘a’ are not equivalent algebraic symbols.)

Sometimes symbols which are characters from other alphabets are used. Mathematicians are particularly fond of characters from the Greek alphabet. In fact, it isn’t too uncommon to use special symbols which are not part of any alphabet. However, regardless of how exotic the symbols used may be, in algebra each symbol still simply represents a numerical value.

Actually, we can distinguish at least three types of literal symbols in algebra:

1) constants – these are symbols which by convention always represent the same numerical value and are used more as a matter of convenience than anything else. For example, the Greek character (pronounced “pie”) is often used to represent the ratio of the circumference of a circle to its diameter. This is an irrational number equal to 3.141592653589793 to fifteen decimal places. It is exactly the same value for every circle. Obviously, it is much easier to write the simple symbol than it is to write a sixteen or more digit approximation to the actual value of .

2) variables – these are symbols that can either represent any numerical value, or which represent only one or a few possible numerical values, but for the moment we don’t know precisely what those values are. So, for example, in the formula for the area, A, of a circle of radius, r

the symbol represents a constant, as explained earlier. However, the symbol r represents the actual radius of the circle whose area you wish to calculate. The actual value of r will be different when circles of different sizes are being considered, and in principle, r could have any positive value. Similarly, the symbol A represents the area of the circle under consideration. In principle, the value represented by A could also be any non-negative number. Of course, once you focus on a specific circle with a specific value of r, the specific value that the symbol A represents is fixed by this formula.

3) arithmetic operations and relations – we don’t usually think of these symbols as algebraic symbols, but they are present in algebra and contribute essential information. At a basic algebra level, there are just a few of these:

 + - “minus” or “subtract” × “times” or “multiply” ÷, / “divide by” = “equals,” indicates that the quantities or expressions to the left and right represent the same numerical value.

Numbers or symbols as superscripts indicate powers or exponents as with ordinary numerical arithmetic expressions.

Generally these arithmetic operation symbols must always be explicitly present. The one exception is that the multiplication of quantities represented by two symbols may be represented by just writing the symbols adjacent to each other with no space in between. Thus

ab often means a × b

and

3a often means 3 × a

(This can be ambiguous, since it is often convenient to use multi-character symbols. For example, if we wrote the formula for the area of a circle as

it is clear that the right-hand side means “multiply times r 2 ”. However, the symbol “Area” on the left-hand side is a single symbol for the value of the area of the circle, and does not mean “A times r time e times a.” In particular, this rule that adjacent symbols means multiplication is never used for two or more numerical digits adjacent to each other. So “37” always means “thirtyseven” and never “3 times 7.” To indicate “3 times 7,” you would need to write “3 × 7” or (3)(7).)

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