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Math for a Digital Age

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Title: Math for a Digital Age


1
Math for a Digital Age
2
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. 

3
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.

4
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.

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.

6
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.

7
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.

8
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.

9
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.

10
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.

11
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.

12
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 0 x 4 0 x 2 1 x 1 128 16 1 145

13
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.

14
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 numbers 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.

15
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)

16
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.

17
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.

18
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 0 R 2
  • The result is 2322 in octal.

19
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
  • 283 382 281 280 1024 192 16 2
    1234.
  • Any number raised to the power of zero is one.

20
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.

21
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.

22
Laboratory Safety and Tools
23
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.

24
Workspace Practices that Help Reduce ESD
potential
  • A wrist strap is a device that is attached to the
    technicians 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.

25
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 characteristicusually 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.

26
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.

27
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

28
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

29
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.

30
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.

31
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.

32
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