Prependix A: Math Notation On KCT

The Problem

Parentheses, Brackets, and Order of Operations

I will use parentheses and brackets in the standard way, to specify the order of operation in arithmetic expressions. I will use parentheses in preference to brackets for this purpose. When expressions use only one line of typed text, parentheses will simply show as in ( expression ). When expressions take more than one line, you will see something like:

a + b c + d

+ n

Wherever parentheses appear, they indicate that the operations contained inside the parentheses should be done before those outside of the parentheses.

I shall use brackets primarily to indicate subscripts. This is in line with how computer program languages indicate subscripting. So an expression of A[13] can be read as A sub 13. When it is not confusing, I shall use A₁₃ to mean the same thing. Sometimes (and only when it is unavoidable) I will use brackets to indicate grouping of operations. Typically they will be used for higher groupings than the parentheses. So you might see something like:

a + b c + d

+ n

× m

When no parentheses are shown, you should do arithmetic operations in the following precedence: exponentiation first, then multiplication, then division, the addition and subtraction.

Addition and Subtraction

I will use the ordinary symbols: + for addition and - for subtraction. So a + b means a plus b, and a - b means a minus b. I assume that you already know that addition is commutative (that is a + b = b + a) and associative (that is (a + b) + c = a + (b + c)).

Multiplication

Multiplication is rendered either with or without using the "×" symbol. The equation,  a × (b + c) = (a × b) + (a × c)  is rendered more succinctly by  a(b + c) = ab + ac , which is the more traditional notation. When no operator symbol is shown between two expressions you should assume that the two expressions are multiplied.

Fractions and Division

The forward slash character will be used to indicate division. So a / b reads a divided by b or a over b. Quite often I will use a horizontal line to indicate division. So

   a + b
        
   c + d

Exponentiation

Superscripting is used to indicate exponentiationl. So x^y indicates x raised to the y power.

The exponential function, usually denoted as:

   ex

   exp(x)

Absolute Values

The notation |x| denotes the absolute value of x. That means if x is negative, its absolute value is positive, but of the same magnitude. If x is positive or zero, the its absolute value is the same as x.

Radicals

A square root is always the positive square root unless the ± symbol precedes it, as in ±√x. In that case, it means both the positive and negative square roots of x.

For higher order roots, I will use the notation of: ( expression )^1/n, which can be read as the nth root of expression.

Inequalities and Relational Symbols

a > b means a greater than b.
a < b means a less than b.
a ≥ b means a greater than or equal to b
a ≤ b means a less than or equal to b
a ≠ b means a not equal to b

Variables

Most variables symbols will be single letters. Some examples are a, b, i, j, t, u, v, x, y. Sometimes I will use upper case letters as well for variable symbols. Often the upper case letters will be used to represent constants. Variables can represent integers, real numbers, constants, or functions. When the independent variable of a function needs to be shown, it will follow the symbol for the function immediately, but in parentheses. A function, f, that takes an independent variable, x, for example, will be shown as f(x), which you can read as f of x. Functions of more than one independent variable will show the variables set off by commas: f(x,y).

When letters are used to represent sets or vectors, I shall render them in bold type: A, B, u, v, chi, PSI

Some variables in calculus are traditionally shown using Greek letters.

  Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω

  α β γ δ ε ζ η θ ι κ λ μ ν ξ ο π ρ σ τ υ φ χ ψ ω

Ordered Pairs

The standard notation of (x,y) will be used to indicate an ordered pair, which will often mean it is a point on a Cartesian plane. Likewise (x,y,z) is an ordered triple, and might be used to represent a point in Cartesian 3-dimensional space. This notation can be extended to any number of dimensions.

If A is a set and B is a set, then A × B is the Cartesian product of the two sets. That means it is the set of all ordered pairs that you can make by taking the first element of the ordered pair from A and the second from B.

Factorial

The expression, n!, where n is a counting number, indicates the product of all the counting numbers up to and including n, and is called, n factorial. So for example,

   4!  =  1 × 2 × 3 × 4  =  24

Very rarely you will see the expression, n!!. This indicates not the factorial of a factorial, but rather, the product of all the odd counting numbers up to and including n. So

   7!!  =  1 × 3 × 5 × 7  =  105

Using factorial expressions, you can make binomial coefficients. The expression, b(n, k) is the kth binomial coefficient of the nth degree (where k ≤ n). The formula is:

b(n,k) =

n k

=

n! k!(n-k)!

The Sigma Summation Symbol

To show a summation over a series of indexed variables or expression, I shall sometimes use the standard sigma notation,

    n
    ∑  Aj
   j=1

   A1 + A2 + A3 + ... + An

Infinity

The symbol, ∞, indicates infinity. Note that in calculus we never use infinity as a number, but rather as a limit. So if x can range from zero to ∞, then it lower bound is zero, but x has no upper bound.

Limits

The expression:

    lim   f(x)
   x → 0 

Derivatives

I shall be using several standard notations for taking derivatives. The "d" notation is:

     dy
       
     dx

Second derivatives can be indicated in either of two ways:

   d2y
        =  y"
   dx2

  dny
       =  y(n)
  dxn

Partial derivatives are a real problem, since there is nothing that looks even remotely like the standard symbol for that (the standard symbol looks like a backward '6').

    ∂y
      
    ∂x

Integrals

The symbology of:

b a

f(x) dx

reads: the integral from a to b of f(x) dx

Logical Implication Symbols

The symbols for logical implication will always be shown in bold type to distinguish them from similar-looking relational symbols (like less than or equal to).

   statement 1  =>  statement 2

  statement 1  <=  statement 2

  statement 1  <=>  statement 2