Introduction to Digital Electronics

Sources

1. Logic circuits part 1 (Lecture Slides)
2. Logic circuits part 2 (Lecture Slides)

Digital Signals

• In contrast to an analog signal, a digital signal, theoretically, can only have an output limited to two different values. For example, it can only have a voltage of either 5V or 0V, whereas an example analog signal can have voltage multiple values ranging between 5 and 0.
• Three common ways to refer to the two states of digital signals:
  • high and low
  • true and false
  • logic 1 and logic 0

Computers

• The computer is a perfect example of the application of digital signals: they are powerful machines with signals consisting of only two states, a high state and a low one.
• As a result of being digital, the computer language is restricted to 2 characters.

Number System

• The number system is helpful for understanding 2 character languages, like the ones used in computers.
• The radix or base indicate the number of possible unique characters of a particular number language.
  • The most popular one is the base 10—which has 10 unique characters: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
• Arithmetic operations require that the numbers possess the same base; otherwise, it is invalid.

Number System Names Based on their Radix

Name	Base/Radix	Symbols
Binary/Dyadic	2	0,1
Tertiary/Ternary	3	0,1,2
Quarternary/Quatenary	4	0,1,2,3
Quinary/Quintary	5	0,1,2,3,4
Senary	6	0,1,2,3,4,5
Septary/Septenary	7	0,1,2,3,4,5,6
Octal/Octenary	8	0,1,2,3,4,5,6,7
Nonary	9	0,1,2,3,4,5,6,7,8
Decimal	10	0,1,2,3,4,5,6,7,8,9
Undenary	11	0 to 9, A
Duodecimal/Duodenary	12	0 to 9, A to B
Tredenary	13	0 to 9, A to C
Quartuodenary	14	0 to 9, A to D
Quindenary	15	0 to 9, A to E
Hexadecimal	16	0 to 9, A to F
Vicenary/Vigesimal	20
Sexagesimal	60

Binary

• In binary, 4 bits is equivalent to 1 nibble.
• On the other hand, 8 bits is equivalent to 1 byte

Converting to Base 10

To translate a base r number to base 10, each digit should be multiplied by the corresponding power of r. Afterwards, get the sum of the resulting products.

Converting from Base 10 to Base R

1. Divide the number into two parts: the integer component and the fractional component.
2. Convert the integer component into base r
   1. Divide it by r.
   2. Record the quotient and the remainder.
   3. Divide the last recorded quotient by r.
   4. Repeat Step 2 and 3 until you obtain a quotient of 0.
   5. The converted integer component consists of the recorded remainder values; the order should be from the latest to the oldest recorded value.
3. Convert the fractional component into base r
   1. Multiply it by r.
   2. Separately record the integer of the product and the fraction of the product.
   3. Multiply the product’s last recorded fraction by r.
   4. Repeat Step 2 and 3 until your product’s last recorded fraction is 0.
   5. The converted fractional component consists of the recorded product’s integer values; the order should be from the oldest to the latest recorded value.

Converting from Base R1 to Base R2

1. Convert base R1 number to base 10
2. Convert the resulting base 10 number to base R2

TIP: Converting from Octal to Binary

Each octal digit is equal to 3 binary bits.

In this regard, converting from binary bits to octal will begin with grouping the binary bits into 3 digits (starting from the decimal point). Leading 0s may be added to the integer component, while trailing 0s may be added to the fractional component. Lastly, convert each group in a similar way to converting from a binary number to decimal.

Converting from binary to hexadecimal is similar to converting a binary number into octal, except that you group the binary bits into 4 digits, and that values from 10-15 are represented as A to F.