Download Math for a Digital Age

Math for a Digital Age
Measurement-Related
Terminology
• Bit – The smallest unit of data in
a computer. A bit can take the
value of either one or zero, and it
is the binary format in which data
is processed by computers.
• Byte – A byte is used to describe
the size of a data file, the amount
of space on a disk or other
storage medium, or the amount of
data being sent over a network.
One byte consists of eight bits of
data.
• Nibble – A nibble is half a byte or
four bits.
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Measurement-Related
Terminology
Kilobyte (KB) – A kilobyte is 1,024 (or approximately 1,000) bytes.
Kilobytes per second (KBps) – KBps is the amount of data transferred
over a network connection. KBps is a data transfer rate of approximately
1,000 bytes per second.
Kilobit (Kb) – A kilobit is 1,024 (or approximately 1,000) bits.
Kilobits per second (Kbps) – This is the amount of data transferred over
a network connection. Kbps is a data transfer rate of approximately 1,000
bits per second.
Megabyte (MB) – A megabyte is 1,048,576 bytes (or approximately
1,000,000 bytes).
Megabytes per second (MBps) – This is the amount of data transferred
over a network connection. MBps is a data transfer rate of approximately
1,000,000 bytes per second.
Megabits per second (Mbps) – This is the amount of data transferred
over a network connection. Mbps is a data transfer rate of approximately
1,000,000 bits per second.
Measurement-Related
Terminology
• Hertz (Hz) – Hertz is a unit of measurement of
frequency. It is the rate of change in the state or cycle
in a sound wave, alternating current, or other cyclical
waveform. Hertz is synonymous with cycles per second
and it is used to describe the speed of a computer
microprocessor.
• Megahertz (MHz) – One million cycles per second.
This is a common measurement of the speed of a
processing chip.
• Gigahertz (GHz) – One billion (1,000,000,000) cycles
per second. This is a common measurement of the
speed of a processing chip.
Data Representation
• ASCII (American
Standard Code for
Information
Interchange) is the
most widely used
coding scheme to
represent data
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Figure 4-14
Discovering Computers 2010: Living in a Digital World
Chapter 4
5
Analog and Digital Systems
• The world used to depend
entirely on analog
processes, machinery, and
communications for its
functions.
• The variables that
characterize an analog
system may have an infinite
number of values.
• Traditional telephones
transmit voice over copper
wire using analog signals.
Analog and Digital Systems
• In digital systems, the
variables that characterize
them only occupy a fixed
number of discrete values.
• Computers and cable
modems are examples of
digital devices. Digital
devices are gradually
replacing analog devices.
• Digital devices make it
easier to do everyday tasks.
Boolean Logic Gates:
AND, OR, NOT, NOR, XOR
• Computers are built from
various types of electronic
circuits. These circuits
depend on what are called
AND, OR, NOT, and NOR
logic "gates".
• These gates are
characterized by how they
respond to input signals.
Boolean Logic Gates:
AND, OR, NOT, NOR, XOR
• Truth tables” to represent
these statements in a
compact form. Other logic
gate combinations or
extensions such as XOR,
NAND, and so on, are beyond
our scope.
Boolean Logic Gates:
AND, OR, NOT, NOR, XOR
• There are only three primary logic functions: AND,
OR, and NOT.
• The AND gate acts as follows: if either input is off, the
output is off.
• An OR gate acts as follows: if either input is on, the
output is on.
• A NOT gate acts as follows: if the input is on, the
output is off, and vice versa.
• The NOR gate is a combination of the OR and NOT
gates and should not be presented as a primary gate.
• A NOR gate acts as follows: if either input is on, the
output is off.
Decimal and Binary
Number Systems
• The decimal, or Base 10, number system is used every day for
doing math (counting change, measuring, telling time, and so
on). The decimal number system uses 10 digits: 0, 1, 2, 3, 4, 5,
6, 7, 8, and 9.
• The binary, or Base 2, number system uses two digits to
express all numerical quantities. The only digits used in the
binary number system are 0 and 1.
• An example of a binary number is 1001110101000110100101.
Decimal and Binary
Number Systems
• Note that whenever the digit 0 appears on the left side of a
string of digits, it can be removed without changing the string
value. For example, in Base 10, 02947 equals 2947.
• In Base 2, 0001001101 equals 1001101. Sometimes 0s are
include on the left side of a number to emphasize "places" that
would otherwise not be represented.
• Another important concept when working with binary numbers is
the powers of numbers. 20 and 23 are examples of numbers
represented by powers. To describe these examples, say "two to
the zero" and "two to the three". Their values are the following:
20 = 1, 21 = 2, 22 = 2 x 2 = 4, 23 = 2 x 2 x 2 = 8.
• 24 is not equal to 2 x 4 = 8, instead it is equal to 2 x 2 x 2 x 2 =
16.
• There is a pattern. The power is the number of 2s that need to
be multiplied together.
Decimal to Binary
Number Conversions
• The same method is used
with binary numbers and
powers of 2. Look at the
binary number 10010001.
This table can be used to
convert the binary
number 10010001 into
decimal as follows:
• 10010001 = 1 x 128 + 0 x
64 + 0 x 32 + 1 x 16 + 0 x
8+0x4+0x2+1x1=
128 + 16 + 1 = 145
Decimal to Binary
Number Conversions
• To convert a decimal number to binary, the idea is to first find the
biggest power of 2 that will “fit” into the decimal number.
• Consider the decimal number 35.
• What is the greatest power of 2 that fits into 35? Starting with
the largest number, 26, or 64, is too big, so place a “0” in that
column.
• The next largest number, 25, or 32, is smaller than 35. Place a
“1” in that column. Now, calculate how much is left over by
subtracting 32 from 35. The result is 3.
Decimal to Binary
Number Conversions
• Next, ask if 16 (the next lower power of 2) fits into 3.
Because it does not, a “0” is placed in that column.
• The value of the next number is 8 which is larger than 3,
so a “0” is placed in that column too.
• The next value is 4 which is still larger than 3, so it too
receives a “0.”
• The next value is 2 which is smaller than 3. Because 2 fits
into 3, place a “1” in that column. Now subtract 2 from 3,
which results in 1.
• The last number’s value is 1, which fits in the remaining
number left. Thus, place a “1” in the last column.
• The binary equivalent of the decimal number 35 is
0100011. Ignoring first 0, the binary number can be written
as 100011.
The Hexadecimal
Number System
• The Base 16, or hexadecimal,
number system is used frequently
when working with computers, since
it can be used to represent binary
numbers in a more readable form.
• Base 16 uses 16 characters to
express numerical quantities.
• These characters are 0, 1, 2, 3, 4, 5,
6, 7, 8, 9, A, B, C, D, E, and F. An “A”
represents the decimal number 10,
“B” is 11, “C” is 12, “D” is 13, “E” is
14, and “F” is 15. Examples of
hexadecimal numbers are 2A5F,
99901, FFFFFFFF, and EBACD3. A
number such as B23CF
(hexadecimal) = 730063 (decimal)
Binary to Hexadecimal
Conversion
• 1111 in binary is F in hexadecimal.
Also, 11111111 in binary is FF in
hexadecimal.
• When working with these two
number systems, one hexadecimal
character requires 4 “bits,” or 4
binary digits, to be represented in
binary.
• To convert a binary number to
hexadecimal, group the number into
groups of four bits at a time, starting
from the right.
• Convert each group of four bits into
hexadecimal, producing a
hexadecimal equivalent to the
original binary number.
Hexadecimal to Binary
Conversion
• Take each individual hexadecimal
digit and convert it to binary, then
string together the solution.
• Pad each binary representation
with zeros to fill up four binary
places for each hexadecimal
digit.
• The hexadecimal number FE27.
F is 1111, E is 1110, 2 is 10 or
0010, and 7 is 0111. So, in binary,
the answer is 1111 1110 0010
0111, or 1111111000100111.
Converting to Any Base
• If converting from decimal to
octal, Base 8 for example, divide
by 8 successively and keep track
of the remainders starting from
the least significant remainder.
• Take the number 1234 in decimal
and convert it to octal.
• 1234 / 8 = 154 R 2
154 / 8 = 19 R 2
19 / 8 = 2 R 3
2/8=0R2
• The result is 2322 in octal.
Converting to Any Base
• To convert back again, multiply a running total by 8 and add
each digit successively starting with the most significant number.
• 2 * 8 = 16
16 + 3 = 19
19 * 8 = 152
152 + 2 = 154
154 * 8 = 1232
1232 + 2 = 1234
• An easier way of achieving the same results in the above
reverse conversions is by using numerical powers:
• 2*83 + 3*82 + 2*81 + 2*80 = 1024 + 192 + 16 + 2 = 1234.
• Any number raised to the power of zero is one.
Introduction to Algorithms
• An algorithm is a systematic description or method of
exactly how to carry out a series of steps to complete a
certain task. Computers use algorithms in practically every
function they perform. Software is essentially many
algorithms pieced together into a huge set of "code".
• One example already seen is the Euclidean algorithm.
This is essentially the algorithm that is used to do long
division (when dividing two numbers).
• Other algorithm techniques are the number conversion
techniques described previously. The reality is that
vacuuming the carpet or sweeping the garage could both
be algorithms if there is a systematic way that these tasks
are carried out each time. The term does not have to be
used rigidly.
Introduction to Algorithms
• A popular algorithm used by networking devices on
the Internet is the Dijkstra algorithm. This algorithm is
used to find the shortest path between a specific
networking device and all other devices in its "routing
domain". It uses bandwidth as a means of measuring
the shortest path.
• Another common type of algorithm is an encryption
algorithm. These algorithms are used to prevent
hackers from viewing data as it passes through the
Internet. An example is 3DES (pronounced “triple
dez”), an encryption standard used to secure
connections between networking devices and hosts.
Laboratory Safety and Tools
Basic Lab Safety Principles
• The workspace should be
situated away from carpeted
areas because carpets can
cause the build up of
electrostatic charges.
• It should be a nonconductive
surface.
• It should be distant from areas of
heavy electrical equipment or
concentrations of electronics.
• It should be free of dust.
• It should have a filtered air
system to reduce dust and
contaminants.
• Lighting should be adequate to
see small details.
Workspace Practices that
Help Reduce ESD potential
• A wrist strap is a device that is attached to the technician’s
wrist and clipped to the metal system chassis on which the
work is being done.
• Allow 15 seconds to pass before touching any sensitive
electronic components with bare hands.
• A wrist strap can only offer protection from ESD voltages
carried on the body. ESD charges on clothing can still
cause damage.
• Avoid making contact between electronic components and
clothing.
Workspace practices that
Help
Reduce
ESD
potential
• A wrist strap is never worn when working on a monitor or
when working on a computer power supply. Monitors and
power supplies are considered replaceable components.
• Antistatic bags are easily recognized by a shielding
characteristic—usually a silvery-sheen, transparent
appearance. Shielded antistatic bags are important
because they prevent static electricity from entering the
bags.
• When original packaging is not available, circuit boards
and peripherals should be transported in a shielded
antistatic bag. However, never put a shielded antistatic
bag inside a PC.
• If computer components are stored in plastic bins, the bins
should be made of a conductive plastic.
Tools of the Trade
• Most computer repair and
maintenance tools used in the
computer workplace are small hand
tools.
• They are included as part of PC
toolkits that can be purchased at
computer stores.
• If a technician is working on
laptops, then a small torx
screwdriver is necessary.
• The right tools can save a
technician a lot of time and help the
technician avoid damage to the
equipment. Tool kits range widely in
size, quality and price.
Tools of the Trade
The following are workspace organizational aids:
• A parts organizer to keep track of small parts
such as screws and connectors
• Adhesive or masking tape to make labels that
identify parts
• A small notebook to keep track of assembly
and/or troubleshooting steps
• A place for quick references and detailed
troubleshooting guides
• A clipboard for paperwork
Tools of the Trade
The following are some commonly used software tools
in PC computing:
• Partition Magic – Advanced drive partitioning
software
• CheckIt – Fault isolation software
• Spinrite – Hard drive scanning tool
• AmiDiag – Hardware fault isolation software
• DiskSuite – Hard drive defrag software
• SecureCRT – Feature filled terminal software
• VNC – Remote access software
• Norton Antivirus – One of the industry leading virus
protection software
Workspace Cleaning Supplies
• Spray contact cleaner is a
mixture of a solvent and a
lubricant.
• The can usually has a long thin
plastic nozzle inserted into the
head so that it can discharge the
solution in pinpoint fashion.
• Spray contact cleaner is useful
when removing corroded
electrical contacts or loosening
adapter boards with gummy
connection points.
• Do not confuse isopropyl alcohol
with rubbing alcohol.
Workplace Testing Equipment
• A troublesome power source can
cause difficulties for the plugged
in computer system.
• A Fluke 110 Multimeter is used to
test high-voltage devices.
• In addition to the outlet tester and
digital multimeter, wrap plugs
should be part of the standard
equipment kept in the workspace.
• These plugs are also referred to
as loopback plugs, or loopback
connectors.
Lab Safety Agreement
• The Lab Safety Agreement details the procedures to
be followed when working with computers.
• Since many classroom lab exercises will not use high
voltages, electrical safety may not appear to be
important.
• Do not become complacent about electrical safety.
Electricity can injure or cause death.
• Abide by all electrical safety procedures at all times.