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. 

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

Discovering Computers 2010: Living in a Digital World Chapter 4

5Page 221 Figure 4-14

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 + 0 x 4 + 0 x 2 + 1 x 1 = 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 = 0 R 2

• 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 TradeThe 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.