Download PG Reading 01_basicMaths pdf

Communications Engineering MSc - Preliminary Reading
1
What You Should Know About… Basic Maths
This chapter contains a very brief summary of the essential mathematical principles required
before starting on the MSc. It’s not an attempt to teach any material for the first time, it’s a
summary of the essential mathematical ideas you’ll need before you can read any of the other
“Getting Started With…” notes. If you find anything new in this material, you are strongly
advised to revise some basic mathematical methods.
1.1 Functions of Variables
Most of the functions of variables we’ll meet in this course will be functions of time (for
example, an audio signal), delay (for example, the impulse response of a communications
channel) or frequency (for example, the response of a filter). The notation x(t) indicates that
the value of x is a function of the variable t only. x(t,ω) implies that the value of x is a function
of both t and ω.
1.2 Logarithms
The logarithm of a number x is the power to which a given number (known as the base) must
be raised to give x.
baselog( x ) = x
For example, the logarithm (base 10) of 1000 is 3, in mathematical notation log10(1000) = 3,
because 103 = 1000 .
For another example, the logarithm (base 2) of 64 is 6, in mathematical notation log2(64) = 6,
because 26 = 64 .
The sum of the logarithms of two numbers is the logarithm of the product of the two numbers,
since:
baselog( x ) + log( y ) = baselog( x ) baselog( y ) = xy = baselog( xy )
The difference between the logarithms of two numbers is the logarithm of the ratio of the two
numbers, since:
baselog( x ) −log( y ) = baselog( x ) base − log( y ) = baselog( x )
1
base
log( y )
=
x
= baselog( x / y )
y
If you know the logarithm base A of a number x, and you need the logarithm base B, then:
log A ( x) =
log B ( x)
log B ( A)
If you know the logarithm base A of a number x, and you need the logarithm of xa, then:
© 2006 University of York
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Communications Engineering MSc - Preliminary Reading
(
Alog A ( x ) = x a = ( x ) = Alog A ( x )
a
a
)
a
= Aa log A ( x )
log A ( x a ) = a log A ( x)
If any of this is unfamiliar – revise logarithms.
1.3 Algebra
Algebra is an essential tool in engineering – in particular the ability to simplify expressions,
and to express formulas in terms of whichever variables are of most interest. You should be
familiar with results such as:
(x
2
)
− y 2 = ( x − y )( x + y )
and be able to derive results such as:
(x
3
)
(
− y 3 = ( x − y ) x 2 + xy + y 2
)
by division, for example:
x-y
x2 + xy + y2
x3
- y3
x3 – x2y
+ x2y
+ x2y – xy2
xy2 – y3
xy2 – y3
0
You should be able to express formulas in terms of any variable, for example, to change the
formula:
Q=
M
(n − 1) + D + Bn
into:
n=
© 2006 University of York
M − Q( D − 1)
Q( B + 1)
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Communications Engineering MSc - Preliminary Reading
1.3.1
Quadratic Equations
A particularly important result is the formula for quadratic equations:
ax 2 + bx + c = 0
x=
−b ± b 2 − 4ac
2a
and it’s useful to be able to spot the factors of simple quadratic equations: for example, if given
an equation of the form:
x2 − x − 6 = 0
to be able to solve this by inspection, noting that:
x 2 − x − 6 = ( x − 3)( x + 2)
and hence that this is solved by x = 3 or –2.
1.4 Problems
1) If x3 − 7 x 2 + 6 x = 0 , what are the possible values of x?
2) Given the formula: 3n =
2
+ 4 , solve for all possible values of n.
n −1
3) What is logx(x2) ?
4) What is the result of dividing x3 + y 3 by x + y ?
5) Two numbers when multiplied together give 31.25, but when added together give 15. What
are the two numbers?
6) If logx(125) = 3, what is x?
7) If loga(x) = 4, and loga(y) is 2, what is loga(xy)?
8) If (1.05 ) = 2 , take the logarithm of both sides of this equation, and hence solve for x.
x
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