I1100 Computer Science: Number Systems

• The lecture is titled I1100, Introduction to Computer Science, Semester 1, Academic Year 2023/2024, with 3 credits for 30 hours.
• Computer architecture and number systems will be covered in the lecture.

Number Systems Outline

• Definitions

Positional Number Systems

• Decimal system

• Binary system

• Hexadecimal system

• Octal system

• Comparison of numbers in the four systems

• Nonpositional number systems

• One coin has 2 possibilities, heads or tails, or 0 or 1.

• Two coins have 4 possibilities.

• Three coins have 222=8 possibilities.

• Four coins have 2^4=16 possibilities.

Introduction to Number Systems

• A number system defines how a number is represented using distinct symbols.
• A number can be represented differently in different systems
• Example: (2A)16 = (52)8 = (42)10
• Characters are used to create words, but the number of symbols is limited, so repetition and combination are crucial to create words. There are a limited number of digits to show the number
• Digits need to be repeated.
• Number systems are divided into 2 categories: positional and non-positional.

Positional Number Systems

• In positional number systems, the position a symbol occupies in the number determines its value.
• A number in this system can be represented as ±(Sk-1... S2S1S0 . S-1S-2… S-l)b
• Where b is the base.

Decimal System (Base 10)

• There are 10 symbols to represent a number: S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
• The plus sign (+) can indicate the number's positivity, while the minus sign (-) indicates the number's negativity.
• Signs are not stored in computer.
• Real numbers in a decimal system are represented as: ±(Sk-1... S2S1S0 . S-1S-2 … S-l) 10
• The value of an integer (a number with no fractional part) is: n = ±Sk−1 × 10k−1 + ... + S2 × 102 + S1 × 101 + S0 × 10°.
• An example of place values are 274 = 2x10^2 + 7x10^1 + 4x10^0
• Example decimal number 15.2 = 1x10^1 + 5x10^0 + 2x10^-1

Binary System (Base 2)

• The word 'binary' is derived from the Latin root 'bini' (or two by two).
• The system has only 2 symbols: S = {0, 1}
• Symbols are referred to binary digits or bits.
• Data and programs are stored in the computer using a string of bits instead of the other systems because computers have electronic switches with 2 states: on and off.
• We can represent an integer as ±(Sk-1... S2S1S0)2
• The value of an integer (a number with no fractional part) is found using the formula n = ±Sk-1 × 2k-1 + ... + S2 × 2² + S1 × 2¹ + S0 × 2⁰..
• An real number with fractional part is also familiar represented by ±(Sk-1 ... S2 S1 So . S−1S−2… S−l)2
• Real number example: (101.11)₂ equals 5.75 in decimal = 1x4 + 0x2 + 1x1 + 1 x 2^-1 + 1 x 2^-2 = 4 + 0 + 1 + 0.5 + 0.25 = 5.75
• Example: (11001)₂ in binary equals 25 in decimal. n= 1 x 2^4 + 1 x 2^3 + 0 x 2^2 + 0 x 2^1 + 1 x 2^0 = 16 + 8 + 0 + 0 + 1
• The maximum value of a binary integer with k digits is when all digits equal to 1 where nmax=2k-1
• if k = 5, n=(11111)2 = 2^5-1=(31)10

Hexadecimal System (Base 16)

• Binary system is used in computers, but it is inconvenient for representing numbers outside the computer while hex is.
• There's an obvious relation between binary and hexadecimal systems (as well as with the octal system).
• There are 16 symbols with S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}
• A, B, C, D, E, F are equivalent to 10, 11, 12, 13, 14, and 15 respectively.
• Symbols are known as hexadecimal digits.
• An integer can be represented as ±(Sk-1... S2S1S0)16.
• The value of the integer is found with n = ±Sk−1 × 16k−1 + ... + S2 × 16² + S1 × 16¹ + S0 × 16⁰.
• For example, (2AE)16 in hexadecimal translates to 686 in decimal, i.e., 2 x 16^2 + 10 x 16^1 + 14 x 16^0.
• The maximum value of an integer of hex is n^max = 16k-1.
• For example, if k = 5, n=16^5-1=(1048 575)10
• Representing real numbers in hexadecimal is not common.

Octal System (Base 8)

• Devised to show the equivalent of the binary system outside the computer.
• There are 8 symbols: S = {0, 1, 2, 3, 4, 5, 6, 7}
• The symbols are referred to as octal digits.
• An integer is represented as ±(Sk-1... S2S1S0)8.
• The value of an integer is calculated as n = ±Sk−1 × 8k−1 + … + S2 × 8² + S1 × 8¹ + S0 × 8⁰
• Example: (1256)₈ in octal is the equivalent to 686 in decimal.
• The maximum value of an octal integer with k digits is equal to nmax=8^k−1
• Example: if k = 5, n=8^5-1=(32767)10
• Representing real numbers in octal is not common.

Summary of Four Positional Systems

• Decimal: base 10, symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, Example: 2345.56
• Binary: base 2, symbols 0, 1, Example: (1001.11)₂
• Octal: base 8, symbols 0, 1, 2, 3, 4, 5, 6, 7, Example: (156.23)₈
• Hexadecimal: base 16, symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F Example: (A2C.A1)16

Conversion

• Conversion to an equivalent number is possible in another system.
• Conversion from any base to decimal and from decimal to any base will be shown.
• Conversion from binary to hexadecimal or octal and vice versa will be shown.

Conversion from Base B to Decimal

• Multiply each digit with its place value in the source system and add the results to get the number in the decimal system.
• For Example
• Convert the binary number (110.11)₂ to decimal. (110.11)2 = 6.75
• Convert the hexadecimal number (1A.23)16 to decimal. (1A.23)16 ≈ 26.137
• Convert the octal number (23.17)₈ to decimal. (23.17) ≈ 19.234

Conversion from Decimal to Base B

• We can convert a decimal number to its equivalent in any base need 2 procedures: one for the integral part, and one for the fractional part.
• Converting the integral part is done by repetitive division.
• Converting the fractional part is done by repetitive multiplication.
• To convert 35 in decimal to binary (base 2), divide by 2 repeatedly to get the result 35 = (100011)₂
• To convert 126 in decimal to octal, divide by 8 repeatedly to get the result 126= (176)₈
• To convert 126 in decimal to hexadecimal, divide to to get the result 126= (7E)₁₆
• Convert the decimal number 0.625 to binary:
  • 0. 625 = (0.101)10
• Convert the decimal number 0.634 to octal (with a maximum of 4 digits):
  • 0. 634= (0.5044)8
• Convert the decimal number 178.6 to hexadecimal:
  • 0. 625= (B2.9)16

Number of Digits

• In a base b, the number of digits of an integer are found using k = [log♭ N]
• means the smallest integer greater than or equal to x
• Find the required number of bits in the decimal number 234 in all four systems
  • In decimal k = [log₁₀ 234] = [2.37] = 3
  • In binary k = [log₂ 234] = [7.8] = 8 (because 234=(11101010)₂)
  • In octal k = [log₈ 234] = [2.62] = 3 (because 234=(352)₈)
  • In hexadecimal k = [log₁₆ 234] = [1.96] = 2 (because 234=(EA)₁₆)

Binary – Hexadecimal Conversion

• Four bits in binary are one digit in hexadecimal.
• To show the hexadecimal of the binary number (10011100010)2, arrange the number in 4-bit patterns, left to right. The leftmost patter can have 1 to 4 bits. Change the numbers to hexadecimal. As a result (100 1110 0010)2 = (4E2)16.
• To find the binary equivalent of (24C)16, convert each digit to 4-bit patterns. As a result, the result of (0010 0100 1100)2.

Binary – Octal Conversion

• Three bits in binary is one octal digit.
• To show the octal equivalent of the binary number (101110010)2, arrange the number in 3-bit patterns, left to right.

Hexadecimal – Binary – Octal Conversion

• 3 bits in binary converts to octal digits.
• For example binary 1 00 001 001 110 can be split as 4 1 1 6 and 8 1 E becomes 1 000 001 110

Nonpositional Number Systems

• Nonpositional number systems are not used in computers, and use a limited number of symbols in which each symbol has a value
• The position a symbol occupies in the number normally bears no relation to its value

Nonpositional Number Systems: Roman Numerals

• In Roman Numerals S={I, V, X, L, C, D, M}
• Value is 1,5,10,50,100,500 and 1000 respictively
• When a symbol with smaller value is placed after a symbol having an equal or larger value, the values are added.
• Example: III → 1+1+1 = 3
• When a symbol with smaller value is placed before a symbol having a larger value, the smaller value is subtracted from the larger one.
• Example: IV → 5-1 = 4
• A symbol S₁ cannot come before another symbol S₂ if S₁≤10* S₂ Example: I or V cannot come before C.
• Examples
  • VIII →5+1+1+1=8
  • XVIII →10+5+1+1+1=18
  • XIX →10+(10-1)=19
  • LXXII →50+10+10+1+1=72
  • CI →100+1=101
  • MMVII →1000+1000+5+1+1=2007
  • MDC →1000+500+100=1600

Review Questions

1. What is a number system?
2. What is a binary system and what is its' base?
3. How many bits in the binary system are represented by one digit in the hexadecimal system?
4. How many bits in the binary system are represented by one digit in the octal system?

Multiple Choice Questions

1. What is the base of the hexadecimal number system?
2. When converting a decimal integer to base b, do we repeatedly divide or multiply by b?
3. When converting a decimal fraction to base b, do we repeatedly divide or multiply by b?
4. Which of the following representations is erroneous?
   • a. (10111)₂, b. (349)₈, c. (3AB)₁₆, d. 256
5. Which of the following representations is erroneous?
   • a. (10211), b. (342)₈, c. (EEE)₁₆, d. 145
6. Which of the following representations is erroneous?
   • a. (111)₂, b. (346)₈,c. (EEG)₁₆, d. 221
7. Which of the following is equivalent to 12 in decimal? a. (1110)₂, b. (15)₈,c. (C)₁₆, d. None of the above
8. Which of the following is equivalent to 24 in decimal? a. (11000)₂), b. (31)₈, c. (1A)₁₆, d. None of the above

Exercises

1. Convert the binary number (011110.01)₂ to decimal.
2. Convert the hexadecimal number (35E.E1)₁₆ to decimal.
3. Convert the octal number (21.11)₈ to decimal.
4. Convert the decimal number 14.56 to binary.
5. Convert the decimal number 72.8 to octal.
6. Convert the decimal number 16.5 to hexadecimal.
7. Convert the octal number (13.7)₈ to hexadecimal.
8. Convert the hexadecimal number (ABC.D)₁₆ to octal.