Math for a Digital Age

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Math for a Digital Age
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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.

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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.

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Data Representation

• ASCII (American Standard Code for Information
  Interchange) is the most widely used coding
  scheme to represent data

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Discovering Computers 2010 Living in a Digital
World Chapter 4

• Page 221
• Figure 4-14

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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

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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.

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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.

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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)

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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.

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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.

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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.

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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.

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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.

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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.

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Laboratory Safety and Tools
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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.

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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.

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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.

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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.

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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

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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

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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.

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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.

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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.

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