Download MP212 Principles of Audio Technology II

MP212
Principles of Audio Technology II
Review of Math
Derived from Introduction to Technical Mathematics by Peter Kuhfittig
Version 1.0 David Moulton, August 1990
Version 2.0, 11/27/06 – revised JMC
The following material is derived from several chapters of Introduction to Technical Mathematics by Peter Kuhfittig. It is
intended to provide a brief review guide and explanation of some basic material about algebra. If further depth, and or
sample problems and examples are needed, we refer you to the tutor for the course, and more complete materials from
which this material was drawn on reserve in the library.
Copyright © 2006 – Berklee College of Music. All rights reserved.
Acrobat Reader 6.0 or higher required
http://classes.berklee.edu/mpe
MP212 Review of Math
About numb ers:
Most numbers are called rational. This means that they can be expressed as a ratio of any two
integers. Such numbers, when expressed in decimal form exhibit repeating decimal
expansion:
1/3 = .33333333…
1/11 = .090909…
Another group of numbers are called irrational. These numbers cannot be expressed as a ratio
of any two integers. Such numbers, when expressed in decimal form exhibit an endless
nonrepeating sequential expansion:
π = 3.141592654 . . .
square root of two = 1.141592654...
Negative numbers are a concept related to geometry.
If we assume a point in space is defined as 0, and we use numbers to express magnitudes of
distance along a line away from that point, in one direction along the line passing through
the point the numbers will be positive and in the other direction the same sequence of
numbers will occur in mirror image as negative numbers.
For each positive number along an infinitely long line passing through the point 0 there is a
corresponding negative number. The magnitudes of each corresponding positive and
negative number are equal, so that A + (-A) = 0.
The chart below shows the hierarchy of real numbers:
The symbols < and > represent the concepts of "less than" and "greater than" so that:
1 < 5 < 10 > 9
These symbols can be combined with the equality (=) symbol to express the concept of
"equal or greater than" and "equal or less than": < and >.
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The four fundamental arithmetic operations (addition, subtraction, multiplication and
division) performed on numbers are subject to the rules called the commutative, associative
and distributive laws.
The commutative law holds that the sum or product of any pair of numbers is independent of
the order in which they are added or multiplied:
3 + 2 = 2 + 3 and 6 * 7 = 7 * 6
(NB: the symbol "*" will represent multiplication in this handout.)
The associative law holds that the sum or product of any group of numbers added or
multiplied is independent of the grouping:
2 + (3 + 4) = (2 + 3) + 4 and 3 * (4 * 5) = (3 * 4) * 5.
However, when operations are mixed, the operation must be performed within the
parentheses first:
2 + (3 * 4) ≠ (2 + 3) * 4.
The distributive law holds that the product of one number and the sum of two others is equal
to the sum of the products of the first number with each of the others:
5 * (6 + 7) = (5 * 6) + (5 * 7).
Summary of laws:
Addition
Multiplication
Commutative Law
a+b=b+a
ab = ba
Associative Law
(a + b) + c = a + (b + c)
(ab)c = a(bc)
Distributive Law
a(b + c) = ab + ac
(b + c)a = ba + bc
The four fundamental operations are also valid for negative numbers.
The absolute value of any number is equivalent to its difference from zero. The absolute value
for any positive number is the number itself. The absolute value for any negative number is its
corresponding positive number. The absolute value of 0 is 0.
When working with numbers both negative and positive (i.e. with mixed signs) the following
rules hold:
1. To add two numbers with similar signs, find the sum of their absolute values and attach
their common sign: (+4) + (+5) = (+9).
2. To add two numbers with unlike signs, find the difference between their absolute values
and attach the sign of the number with the larger absolute value:
(-4) - (+3) = ?
The difference between their absolute values is 1. The larger absolute value is -.
Therefore the answer is -1.
3. To subtract one number from another, change the sign of the number to be subtracted and
proceed as in addition: 8 - 6 = +8 + (-6) = 2.
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4. The product or quotient of two numbers with common signs is the product or quotient of
their absolute values: +2 * +3 = +6 and -4 / -2 = -2.
The product or quotient of two numbers with unlike signs is the corresponding negative of
the product or quotient of their absolute values: -3 * 4 = -12.
When a number is multiplied by itself, the result can be expressed by the use of an exponent:
3 * 3 = 32 and 2 * 2 * 2 = 23.
The number being multiplied is called the base and the number of times it is multiplied is
called the exponent.
Order of Operations
Multiplication and division are performed first, unless parentheses indicate groupings, in
which case operations within parentheses are performed first.
1. Evaluate (solve for) quantities enclosed in parentheses or other symbols of
grouping.
2. Evaluate powers (exponents)
3. Perform all multiplications and divisions in the order in which they occur from left to
right.
4. Perform all additions and subtractions in the order in which they occur from left to
right.
Operations with Zero
a+0=a
a–0=a
a*0=0
0/a=0
Any number divided by 0 cannot be determined (is indeterminate). Therefore, division by 0 is
not a viable arithmetic operation.
Letters used to represent numbers are called literal symbols. If such a symbol can take on
various values, it is called a variable. A number or literal symbol that does not vary is called a
constant. Literal symbols occur in formulas.
A formula expresses a relationship among variables. Ohm's Law, for instance, states that Volts
(E) = Current (I) * Resistance (R), or:
E = IR.
If the resistance R is fixed (a constant) then E will vary as a function of the variation of I. Please
note also the convention that two symbols next two each other (IR) represent the product of:
I * R.
More about grouping:
In algebra, terms are often grouped to indicate that they are to be considered as a single
quantity or number. Often, in algebra, we need to remove symbols of grouping, as follows:
5(a + 3(a + b)) =
5(a + 3a + 3b) =
5(4a + 3b) =
20a + 15b.
When removing groupings, work from the innermost grouping outwards.
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Equations
An equation is an equality between two expressions:
1. 1/2 = 2/4
2. x = 1
3. 2y + 3y = 5y
4. 3x + 1 = 4
Equation 3 is valid for all numbers y and is called an identity.
Equation 4 is only valid for x = 1 and is called a conditional equation.
The word equation is commonly restricted to refer to conditional equations.
The literal symbol in a conditional equation is called the variable, which may represent some
unknown quantity. The value of the unknown for which the equality holds is called the
solution and is said to satisfy the equation.
The most direct way to solve an equation is to make use of the fact that both sides of the
equation are numbers, and that they are equal numbers. The equality can then be preserved
throughout a series of algebraic manipulations if we perform the same operations on each
side of the equation, or:
the equality can be preserved if we add the same number to both sides of the equation,
subtract the same number from both sides, multiply both sides by the same number, or
divide both sides by the same number.
The purpose of the above manipulations is to bring the unknown symbol to stand alone on
one side of the equality, so that it can be seen in equality with numbers or other symbols, as
in:
x - 14 = 3.
Therefore, in our quest to get x to stand alone, if we add 14 to both sides, we get:
x - 14 + 14 = 3 + 14 =
x = 17.
Or, in another instance,
2x = 5.
If we divide both sides by 2, we get:
(2x)/2 = (5)/2 =
x = 2.5.
Another example:
2x - 12 = 5.
If we divide both sides by 2, we get:
(2x - 12)/2 = (5/2)
(2x/2) - (12/2) = 5/2
x - 6 = 2.5.
If we add 6 to both sides, we get:
x - 6 + 6 = 2.5 + 6 =
x = 8.5
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A formula is an equation that is solved for one unknown in terms of others. A formula is a
relationship between variables that often expresses a geometric property or a physical law.
Ohm's Law, again, expresses the relationship between Voltage (E), Current (I) and Resistance
(R):
E=IR
The equation can be expressed in terms of any of its unknowns:
I = E/R
and
R = E/I.
These are all part of Ohm's Law, and constitute re-expressions of it.
Another example is the relationship of Fahrenheit and Celsius temperature scales:
C = 5/9 (F - 32)
To express this in F:
9/5C = 9/5 (5/9(F - 32)) =
9/5C = F – 32 =
9/5C + 32 = F - 32 + 32 = F =
F = 9/5C + 32.
Ratio and Proportion
A ratio is a quotient of two quantities:
1/2 or 36/14 or a/b
Ratios and their expressions are common. For instance, velocity is expressed in miles per hour:
55 miles / 1 hour.
25 miles / 1 gallon is another ratio.
6:1 odds are another ratio.
π is the quotient of the ratio: 1 circumference/1 diameter.
Proportion is an equality of two ratios:
a/b = c/d
2/3 = 8/12
In many cases, proportions are useful because when a known proportion exists and one ratio
is known, the other can be found if either part of its ratio is known.
Variation
A particular type of relationship between numbers is called variation. Direct variation is really
a proportion.
If two variables x and y are related so that y = kx, then y is said to vary directly as x or to be
directly proportional to x. The constant k is called the constant of proportionality.
Given that y = kx, x = 2 and y = 4, then k = 2 and y =2x.
If y is directly proportional to the reciprocal of x (1/x), then y = k(1/x) or y = k/x. This
relationship is called an inverse variation, and we say that y is inversely proportional to x.
If z = kxy, we say that z varies jointly (joint variation) as x and y or that z is directly proportional
to the product of x and y.
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Functions
An important relationship between variable quantities is that of the function. If two variables
x and y are related so that for every value of the variable x there corresponds a single, unique
value of the variable y, then we call y a function of x. X is the independent variable and y is
the dependent variable.
The main purpose of the function concept is to describe an operation on the independent
variable that yields a unique value of the dependent variable. The convention for notation of
functions is:
if y is a function of x, we write y = f(x), which is read to mean y equals f of x.
Therefore, y = x2 can be expressed as f(x) = x2 and so on. The expression f(x) is therefore often
logically assumed.
The Cartesian coordinate system
Invented by Renee DesCartes in the l7th Century, this graphical representation of numbers is
a powerful visual representation of the relationship between two variables, giving humans
rapid right-brain access to large quantities of related numerical data.
The system is based upon the notion of numbers along a line discussed earlier. 0 is a point
(the origin) on a plane (usually a piece of paper). A horizontal line (called the y-axis) through
the zero point is numbered with negative numbers to the left and positive numbers to the
right. A vertical line (called the x-axis) through the zero point is also drawn, with negative
numbers below the origin and positive numbers above it. The resulting drawing is divided
into four zones called quadrants, with upper right (positive y, positive x) being quadrant I, and
the numbering of the quadrants proceed counter-clockwise.
Points on the plane are located by describing the numerical values and directions (or
polarities) from the coordinate axes. The distance from the y-axis is called the x-coordinate, or
abscissa. The distance from the x-axis is called the y-coordinate, or ordinate. To describe a
point, we place the coordinates in parentheses, with the x-coordinate first, so the P is (3,5), Q
is (-4, 2) and R is (-2, -6) in the drawing below.
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The real usefulness of the concept occurs when we combine the concepts of functions and
coordinate mapping, because we get very clear visual templates of the character of a function
using these maps or graphs.
If a function is given by the equation type y = f(x), we assign various values to the
independent variable x and compute the corresponding dependent variable y values. We
then plot these points within the coordinate system, obtaining a graphic representation of
the function.
Formally speaking, the graph of a function y = f(x) consists of all points (x, y) whose
coordinates satisfy the equation y = f(x).
Consider the function y = x + 3.
If x = -5, then y = -2;
If x = -4, then y = -1;
If x = -3, then y = 0;
If x = -2, then y = 1;
If x = -l, then y = 2;
If x = 0, then y = 3;
If x = l, then y = 4;
If x = 2, then y = 5;
If x = 3, then y = 6…
This collection of expressions is called a table and is usually shown as an abbreviated list, such
as:
x: -5 -4 -3 -2 -1 0 1 2 3
y -2 –1 0 1 2 3 4 5 6
and subsequently plotted on the coordinate plane:
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The above example is called a linear function, since the graph representing it results in
a straight line. In general, the form of a linear function is:
f(x) = ax + b, where a and b are constants.
Points where a graph crosses the coordinate axes are called intercepts or zero
crossings, and will be of conceptual importance in audio.
Factoring
Factoring is the process of multiplication in reverse: moving from a known product to its
factors. This process is useful for the manipulation of fractions. It involves the extraction of
common factors from a set of mixed products:
5x + l0y = 20 =
5(x + 2y) = 20 =
(5(x + 2y)/5 = 20/5 =
x + 2y = 4
Factoring can be used to manipulate a fraction to its lowest, or other terms:
25/30 = (25/5)/(30/5) = 5/6
Fractions
The rules for multiplication and division of fractions carry over from arithmetic:
a/b * c/d = ac/bd
1/4 * 2/3 = (1 * 2)/(4 * 3) = 2/12
Division of a fraction is equivalent to the multiplication of the reciprocal of the denominator:
a/b divided by c/d = a/b x d/c = ad/bc
1/4 divided by 2/3 = (1 x 3)/(4 x 2) = 3/8
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When dealing with the addition and subtraction of fractions, it is necessary to make the
denominators of the fractions to be added or subtracted the same, usually by finding the
lowest common denominator (LCD). This unit is defined as:
an expression that is divisible by every denominator and that does not have any more
factors than are needed to satisfy this condition.
The LCD can be found by factoring each of the denominators, then finding the product of the
factors of the denominators.
To add or subtract fractions, we find the LCD for the set of fractions to be added or
subtracted, convert each fraction in the set to an equivalent fraction with that LCD, and then
add or subtract the numerators of the set as needed, and place that sum/difference over the
LCD. We may then simplify the result if needed and possible.
For example:
1/4 and 2/3 have an LCD of 12. We convert 1/4 to 3/12 and 2/3 to 8/12, add the
numerators 3 and 8 and obtain the result 11/12. The conversion of the fraction to its
equivalent is accomplished by multiplying the numerator and the denominator by the
same quantity.
Another example:
(1/2) + (2/5) - (3/4) + (9/8)
Factoring and then multiplying yields an LCD of 40, so that:
((1*20)/(2*20)) + ((2*8)/(5*8)) - ((3*10)/(4*10)) + ((9*5)/(8*5)) =
(20/40) + (16/40) - (30/40) + (45/40) =
51/40
Another example:
(3/x) + (y/2d) =
((3*2d)/(x*2d) + (y*x)/(2d*x)) =
(6d/2dx) + (xy/2dx) =
(6d + xy)/2dx
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