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Seneca College
Math Review
BTP500
Math Notation
49  7

square root,

cube root,

exponents, an = a  a  …  a (n times) “a to the power of n”; 34 = 81, 31 = 3, 30 = 1
3
a n 
125  5
1
an
; fourth root,
4
81  3 , …
2-3 = 18 , 10-6 = 0.000001, 5-1 = 1/5 = 0.2
1
1
1
a n  n a 49 2  7 125 3  5
m
1
3
1
25 2  125
a n  (a m ) n  (a n ) m
ax
ab
1
814  3
100 0.5  10
2
64 3  16
3 2  4.729 (3 decimals)
23  512 but (23 ) 2  64
is also defined for any x, eg
c
means
a (b
c
)
eg:
2

logarithms, inverse operation of powers
logax “log to the base a of x”; logax = y is the same as ay = x
log101000000 = 6, log21024 = 10, log 1 = 0, logaa = 1
on a calculator “log” means log to the base 10; log1000 = 3
on a calculator “ln” means “natural log” to the base e = 2.718; ln 5 = 1.609
in computer science, log usually means to the base 2, NOT base 10

 notation
b
 f (i)  f (a)  f (a  1)  f (a  2)  ...  f (b)
i a
4
example:
i
3
 23  33  43  8  27  64  99
i 2



subscripts
xi means the i-th item
 = “for all”

(x+y)2 = x2 + 2xy + y2 is intended to mean it is true for all values of x and y

so write, more precisely as: x y (x+y)2 = x2 + 2xy + y2

note that x y 2x + 2y = 2x+y is false (try x=1 y=2)
 = “there exists” (or “for some”)

2x + 3 = 11 is intended to mean “solve for x”: there exists an x such that it is true

so write, more precisely as: x 2x + 3 = 11 (which is true: x=5)

note that x x2 = -1 is false (assuming that x is a “real” number)
eg: the average of the n numbers: x1, x2, … , xn is (x1+x2+…+xn)/n =

n
i 1 i
x



mod 
x mod m
means the remainder after dividing x by m, eg: 25 mod 7 = 4
x  y mod m means that x mod m = y mod m, ie x and y have the same remainder
!
n! “n factorial” = number of permutations of n objects
O( ) “order of”
f(n) is O(g(n)) means that f(n) is approximated by some multiple of g(n) when n is large

eg: 5n3 + 100n2 + 88n + 11 is O(n3); because 5n3 is the largest term

eg: 10(2n)+ n55 + n is O(2n)


Lew Baxter
approximately equal, eg 210  103
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Seneca College
Math Review
BTP500
Algebra
m n
am
 a mn
n
a

a a a

log( xy)  log x  log y

log a x 

Sum of arithmetic series
m
n
log x
log a
(a m ) n  a mn
x
log( )  log x  log y
y
eg: log21000000 = (log 1000000)/(log 2)=19.932 using a calculator
n
i 
i 1
n
Sum of geometric series

log x n  n log x
2
i
1
2
n(n  1)
eg: 1+2+3+…+100 = 5050
 2 n1 eg: 1+2+4+8+…+512 = 1023
(note the binary: 1111111111)
i 0
Solving Equations
5x – 7 = 13  5x = 20  x = 4
x2 = 7  x = 7 = 2.646; generally ax2 + bx + c = 0
x3 = 100  x =
3
100

x
 b  b 2  4ac
2a
= 4.642 (or use inverse power “inv ^” on a calculator)
x
3 = 88  x log 3 = log 88  x = log 88 / log 3 = 4.075
Functions
constant f(x) = c eg: f(x) = 5
linear f(x) = ax+b eg: f(x) = 3x + 7
quadratic contains x2, x eg: 3x2 + 5x +1, cubic contains x3, x2, x eg: 7x3 + x2 + 4x + 1
polynomial contains only powers of x eg: 5x7 + x3 + 20x, x100, x2 + 5x, 3x + 7, 20
exponential grows very fast - faster than any polynomial eg: 2n nn n! 1.001n
logarithmic: grows very slowly - slower than 1/polynomial log x
n log n : between linear and quadratic
22
n
Graphs
Websites with math tools
http://pirate.shu.edu/~wachsmut/Java/IRA/Plot.html - plot multiple functions, with zooming
WIMS : http://wims.unice.fr/wims/en_home.html - many online calculators, function plotters
Lew Baxter
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