CSIT 230: Number Systems

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Questions and Answers

Why are number systems essential in the context of computer systems?

  • They are used for advanced audio processing.
  • They manage power distribution within computer hardware.
  • They serve as the foundational language for computers. (correct)
  • They facilitate graphical user interface design.

In digital systems, which number system is predominantly utilized?

  • Octal
  • Decimal
  • Hexadecimal
  • Binary (correct)

What is the primary reason for using hexadecimal representation in computer systems?

  • To reduce the complexity of floating-point arithmetic.
  • To make binary representation more compact and readable. (correct)
  • To enhance security in data transmission.
  • To improve the speed of data processing.

When converting the decimal number 953 to its base-10 representation using powers of 10, which expression accurately represents this conversion?

<p>$9 \cdot 10^2 + 5 \cdot 10^1 + 3 \cdot 10^0$ (D)</p>
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In the context of binary numbers, what does the term 'bit' refer to?

<p>A binary digit (C)</p>
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In a binary number, what does MSB stand for, and what does it represent?

<p>Most Significant Bit, representing the highest value in the number (D)</p>
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Which of the following statements accurately describes a characteristic of binary numbers?

<p>Binary numbers are a base-2 numeral system. (C)</p>
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What is the decimal equivalent of the binary number 1100100?

<p>100 (A)</p>
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What digits are used to represent numbers in the octal system?

<p>0-7 (A)</p>
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What is the representation of the decimal number 8 in octal?

<p>10 (C)</p>
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In the hexadecimal system, what decimal value do the symbols A through F represent?

<p>10-15 (A)</p>
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In hexadecimal, what single digit is used to represent the decimal number 13?

<p>D (D)</p>
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What is the result of subtracting 1 from the base of the hexadecimal system?

<p>F (A)</p>
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What is the decimal equivalent of the hexadecimal number 2F?

<p>47 (C)</p>
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Which process is used to convert a base-10 number into a binary number?

<p>Dividing by 2 (C)</p>
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What is the binary equivalent of the decimal number 27?

<p>11011 (C)</p>
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What is the decimal equivalent of the binary number 1011?

<p>11 (B)</p>
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What arithmetic operation is used to transform an integer to binary, but an integer with a fractional component to binary?

<p>Decimal to Binary conversion, use: Divisions (A)</p>
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What is the initial step in converting a decimal fraction to its binary equivalent?

<p>Multiply the fraction by 2. (A)</p>
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When should the multiplication-by-2 process be stopped when converting a decimal fraction to binary?

<p>When the fraction becomes 0 or the desired precision is reached. (C)</p>
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What is the binary equivalent of the decimal fraction 0.6875?

<p>0.1011 (B)</p>
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How do you combine the integer and fractional binary equivalents after converting a mixed decimal number into binary?

<p>Concatenate them with a decimal point. (C)</p>
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What is the key idea behind converting from binary to octal?

<p>Group the binary digits into sets of three. (D)</p>
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When converting from binary to octal, into what size sets must the binary digits be grouped?

<p>3 (B)</p>
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What is the octal equivalent of the binary number 010011?

<p>23 (D)</p>
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What procedure is used for converting binary numbers into hexadecimal numbers?

<p>Group the binary digits into sets of four. (C)</p>
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When converting from binary to hexadecimal, what is the size into which groups binary digits must be placed?

<p>4 (B)</p>
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What does the process of converting from octal to binary entail?

<p>Representing each octal digit with three binary digits. (A)</p>
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What is the binary equivalent of the octal digit 7?

<p>111 (B)</p>
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What does converting from hexadecimal to binary involve?

<p>Converting each hexadecimal digit to a four-digit binary number. (B)</p>
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Which binary code is used to represent the hexadecimal number D?

<p>1101 (C)</p>
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When dealing with a number that has both an integer and a fractional part, what is the general approach for converting it from decimal to binary?

<p>Convert each part separately and then combine them. (D)</p>
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After converting the decimal number 41.6875 into binary, which result accurately represents its binary equivalent?

<p>(101001.10110)2 (C)</p>
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What is the primary base used in the decimal number system?

<p>10 (D)</p>
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What is the primary characteristic of the binary number system?

<p>It uses only two digits, 0 and 1. (A)</p>
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What is the base of the octal number system?

<p>8 (B)</p>
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Which of the following number systems consists of 16 symbols?

<p>Hexadecimal (D)</p>
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Which of the following is true regarding 1-bit binary numbers and its decimal equivalent?

<p><code>0</code> in binary equals <code>0</code> in decimal, and <code>1</code> in binary equals <code>1</code> in decimal. (C)</p>
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What is the range of a 3-bit binary number?

<p>0 to 7 (D)</p>
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Which of the following represents the decimal number 9 expressed in octal?

<p><code>11</code> (D)</p>
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What symbols does the octal system use to represent numbers?

<p>0-7 (A)</p>
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What would the number 10 in decimal be in octal?

<p>12 (C)</p>
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What decimal number would the symbol 'F' represent in a hexadecimal system?

<p>15 (C)</p>
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What is the maximum single digit in hexadecimal?

<p>F (A)</p>
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What is the result of calculating n-1, where n is the base of the number system, in the octal number system?

<p>7 (C)</p>
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Calculate the result of n-1 using hexadecimal, where n is the base of the number system, convert to decimal, and then give the corresponding hexadecimal digit.

<p>15 which corresponds to F (B)</p>
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To convert a base-10 number to binary, what mathematical operation is primarily used?

<p>Division (A)</p>
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If you have a number with values after the decimal point, what two operations should you use to convert the number to binary?

<p>Division and multiplication (B)</p>
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According to the provided slides, what must you do to convert each digit from binary to decimal?

<p>Multiply each binary digit by the appropriate power of 2 (C)</p>
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What operation is known as the inverse of transforming an integer divided by two?

<p>Multiplication (D)</p>
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How do you convert multiple digits from binary to octal?

<p>Group the digits (C)</p>
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If you start with a binary number, into what size sets must you group the binary numbers to convert them to octal?

<p>Sets of three (A)</p>
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What principle underlies the conversion from hexadecimal to binary?

<p>Each hexadecimal digit is represented by a group of four binary digits. (C)</p>
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Regarding converting from octal to binary, suppose you have a number such as (306.7)_8, what is the resulting operation

<p>(011 000 110.111)_2 (B)</p>
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To convert a base-10 number with both integer and fractional parts into binary, what is the correct approach?

<p>Convert each part, integer and fraction, separately and then combine them. (A)</p>
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To convert decimal, multiplication occurs on the fraction, but how do you treat integers?

<p>Divide (B)</p>
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If you want to convert a number to hexadecimal, into what size sets must you group the digits?

<p>Sets of four (B)</p>
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What is the outcome of applying division to 27?

<p>11011 (A)</p>
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If 28 is divided repeatedly, what are the remainders (in order)?

<p>0 0 1 1 1 (D)</p>
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What is the result of 153 / 8, what is remainder? What is the next calculation?

<p>1, and next calc is 19 / 8 (D)</p>
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What is the overall strategy of converting integer numbers?

<p>Proficiency must be increased to proficiency in changing integer numbers, by dividing. (C)</p>
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What two concepts are introduced while understanding number systems?

<p>Number with fractions, and changing integer numbers (B)</p>
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According to the slides, what is one of the most important concepts as a Key Takeaway?

<p>The language of computers is very important (C)</p>
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Considering the relationship between binary, octal, and hexadecimal number systems, which of the following statements is true?

<p>Octal and hexadecimal number systems both serve as shorthand for binary representation. (D)</p>
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If you were tasked with converting the decimal number 350 into its binary equivalent, which of the following steps would you undertake?

<p>Divide 350 repeatedly by 2 and recording the remainders. (D)</p>
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Given the binary number 101101, which of the following mathematical expressions correctly represents its conversion to decimal?

<p>$1<em>2^5 + 0</em>2^4 + 1<em>2^3 + 1</em>2^2 + 0<em>2^1 + 1</em>2^0$ (B)</p>
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What is the decimal equivalent of the octal number 73?

<p>59 (C)</p>
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You have the hexadecimal number 3E. What are are the steps (in order) required to convert it to decimal?

<p>Convert <code>3E</code> to binary, then convert the binary number to decimal. (A)</p>
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Flashcards

Decimal Number System

The base-10 number system, used in everyday calculations.

Binary Number System

The base-2 number system; the language of computers.

Bit

A binary digit, either 0 or 1.

Octal Number System

Base-8 number system.

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Hexadecimal Number System

Base-16 number system; uses digits 0-9 and letters A-F.

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Most Significant Bit (MSB)

The leftmost bit in a binary number; carries the most weight

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Least Significant Bit (LSB)

The rightmost bit in a binary number; carries the least weight

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Decimal Conversion

A method to change a number from one base to another using powers of 10.

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Decimal to Binary Conversion

A method to convert from another base to base-10 using repeated division.

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Decimal to Octal Conversion

Involves repeated division by the base until the quotient is zero.

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Integer with Fraction

A number that includes both a whole number part and a fractional part.

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Decimal to Binary (with fraction)

Convert the whole and fractional parts separately, then join them.

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Integer Part Conversion

Dividing the integer by the base and writing the remainder.

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Fractional Part Conversion

Multiply the fractional part by 2 multiple times.

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Hexadecimal System

A system that represents numbers in base 16.

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Octal to Binary Conversion

The process of changing a number from octal (base-8) to binary (base-2).

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Hex to Binary Conversion

The process of converting a number from hexadecimal (base-16) to binary (base-2).

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Study Notes

  • The lecture is for CSIT 230 Computer Systems at Montclair State University, taught by Dr. Prince Waqas Khan
  • Number systems and their conversions are discussed

Number Systems

  • Number systems are the language of computers

Common Number Systems

  • Decimal is base 10 and represents our everyday number system
  • Binary is base 2 and is used in digital systems
  • Octal is base 8 and acts as shorthand for binary representation
  • Hexadecimal is base 16 and is a compact representation of binary

Decimal Numbers

  • Decimal Numbers are base 10
  • In base 10, every digit is multiplied by the appropriate power of 10
  • Example: the decimal number 953 = (9 * 10^2) + (5 * 10^1) + (3 * 10^0)

Binary Numbers

  • Binary numbers are base-2 numeral system, using only the digits 0 and 1
  • Each binary digit is called a "bit"
  • The most significant bit (MSB) is the leftmost bit, and the least significant bit (LSB) is the rightmost bit

Octal Numbers

  • The octal system has a base of eight, each place representing a power of eight
  • Octal only uses the digits 0-7
  • The decimal number 8 would be represented as 10 in octal
  • The decimal number 9 would be represented as 11 in octal

Hexadecimal Numbers

  • The 16 symbols used in hexadecimal system are the numbers 0-9 and the letters A-F
  • The symbols 0-9 represent the same values as in the decimal system, while A-F represent the values 10-15
  • The hexadecimal system is also known as base-16
  • In any number system, the highest possible number that can be represented is n-1, where is n is the base of the number

Number Conversions

  • Integer:

    • Binary to Decimal
    • Decimal to Binary
    • Decimal to Octal
    • Decimal to Hex

  • Integer with Fraction:

    • Decimal to Binary
    • Binary to Octal
    • Binary to Hex
    • Octal to Binary
    • Hex to Binary

Converting from Base 2 to Base 10

  • Take (xxxx) in base 2 and convent to (yyyy) in base 10

Converting from base-2 to base-10

  • Multiply each binary digit by the appropriate power of 2
  • Example:
    • (1110) in base 2
    • = (1 * 2^3) + (1 * 2^2) + (1 * 2^1) + (0 * 2^0)
    • = (8) + (4) + (2) + (0)
    • = (14) in base 10

Converting from Base 10 to Base 2

  • Decimal numbers are converted to Binary numbers
  • Binary to decimal conversions use multiplication
  • Decimal to binary conversions use division
  • Repeatedly divide the decimal number by 2, and note the remainders until the result is zero
  • Final Result: The remainders written from bottom to top

Octal Numbers

  • (153) in base 10 is equal to (?) in base 8
  • 153 / 8 = 19 remainder 1
  • 19 / 8 = 2 remainder 3
  • 2 / 8 = 0 remainder 2
  • Final answer: (231) in base 8

Hexadecimal Numbers

  • (245) in base 10 is equal to (?) in base 16
    • 245 / 16 = 15 remainder 5
    • 15 / 16 = 0 remainder 15
    • (F5) in base 16

Integer with a Fraction

  • To convert decimal numbers with both integer and fraction parts into binary, convert each part separately and then combine them
  • Integer Part Conversion:
    • Divide the integer part repeatedly by 2, and noting down the remainder.
    • Write the remainder in reverse order to get the binary equivalent.
  • Fractional Part Conversion
    • Multiply the fractional part by 2 repeatedly.
    • Note down the integer part (0 or 1) after each multiplication.
    • Continue until the fraction becomes 0 or the desired precision is reached

Example Fractional Part Conversion

  • Convert (0.6875) in base 10 into (?) in base 2
  • 0.6875 * 2 = 1.3750
  • 0.3750 * 2 = 0.7500
  • 0.7500 * 2 = 1.5000
  • 0.5000 * 2 = 1.0000
  • 0.0000 * 2 = 0.0000
  • = (0.10110) in base 2

Converting from Binary to Octal

  • Conversion from binary to octal, and vice versa
  • Step 1: Start from A, go left 3 digits
  • Step 2: Start from A, go right 3 digits
  • Example: Given (010 110 001 101 011.111 100 000 010) in base 2, convert to (?) in base 8
  • 011 = 3
  • 101 = 5
  • 001 = 1
  • 110 = 6
  • 010 = 2
  • 111 = 7
  • 100 = 4
  • 000 = 0
  • 010 = 2
  • The final solution : (26153.7402) in base 8

Converting from Binary to Hexadecimal

  • Conversion from binary to hexadecimal, and vice versa
  • Step 1: Start from B, go left 4 digits
  • Step 2: Start from B, go right 4 digits
  • Example, Convert (10 1100 01101011. 11110010) in base 2 into (?) in base 16
  • 0010 = 2 convert
  • 1100 = C
  • 0110 = 6
  • 1011 = B
  • 1111 = F
  • 0010 = 2
  • The final solution: (2C6B.F2) base 16

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