Data Types
What is
represented by
the number:
1101
Binary
Decimal
Octal
Hex
(13)10
(1101)10
(577)10
(4353)10
In a computer, the base
is not ambiguous (it's
always binary), but how
0's and 1's are
interpreted still
depends on the type of
data being represe
Floating Point
Representations and
Scientific Notation
So far we have discussed two data types:
unsigned real numbers (binary expansions)
and signed integers
Representing very large and very small real
numbers as binary expansions is limited by the
number
Canonical Forms
In general there is no unique Boolean
expression corresponding to a given
truth table
Canonical forms make it possible to
write the Boolean expression for a
given truth table in a standardized
way that is unique
The two canonical forms tha
Logic Circuits
1. Determine the input and output variables
Assign variable names and interpret the meaning
of '0' and '1' in each case
2. Build a truth table that shows the desired
output values for each combination of input
values
3. Construct the corres
1000011 1001111
COMP 1113
1001101 1010000
Applied
10001011001
Mathematics
Instructor: Paul Rozman
Office: SW2-229
Phone: 604-451-8718
E-mail: [email protected]
Positional Number Systems
Positional number
systems are number
systems in which the
significance
Boolean Algebra
Variables in a
Boolean algebra take
on values that are
not numbers but
logical values:
TRUE or FALSE
x = TRUE, y = FALSE
George Boole
1815 - 1864
It is not of the essence of mathematics to be conversant with
the ideas of number and quantit
Systems of Linear Equations
Method of Substitution
Expressed in the form, y = f(x), equations must
be linear in x:
1
4 x 3 y 1
6 x 4 y 2
2
Solve by substitution:
y 43 x 13
6 x 4(
2
3
4
3
10
3
1
3
4 x 1
3
) 2
x x 5
y (5) y 7
FROM
1
SUBSTITUTE
IN
SOLVE FOR
COMP 1113 Applied Mathematics for CST
LAB: Signed Integer Systems and Arithmetic Operations in 2s Complement
SIGNED INTEGERS
1. Use an 8-bit representation to write the following decimal numbers in each of the
listed representations, where possible. Indic
Glossary of
Gates
Basic Gates
NOT
gate (inverter)
x
x
AND
gate
x
y
x
y
x* y
OR
Derived Gates
x
y
gate
XOR
gate
x
y
x
y
gate
NOR
x
y
gate
x+y
NAND
xy
x + y
x y
XNOR
gate
x y
Constructing a Logic
Circuit
u'
y
yu'
yz'
badly
yz' + y'zu + yu'
Q
z'
z
Q yz
Power Functions
x3
Power functions
raise a variable x
to a constant
power
E.g. x2 or x10
Observe that if x>
1, then
1<x<x2<x3
x2
x 3/2
x
x 1/2
x 1/8
Exponential Functions
Power functions
raise a variable x
to a constant
power
E.g. x2 or x10
y = 10x
y =
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COMP 1113
Exercises #9: Matrices
1. For each of the following pairs of matrices, calculate whichever of the products AB or BA
which are defined. (In some cases, neither product is defined, in others one but not the other,
and in some both products may be
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COMP 1113 Applied Mathematics for CST
Lab 9 Truth Tables and Boolean Proofs
Use truth tables to prove the following Boolean statements. Show that the same result is
obtained using Venn diagrams:
1. x + x' = 1
2. x*(x + y) = x
3. x*(x' + y) = x*y
4. (x + y