What happens when a smaller symbol is placed before a larger symbol in Roman numerals?
The smaller value is subtracted from the larger one.
p.10
Conversion Between Number Systems
What is done with the remainder during the conversion process?
The remainder is inserted to the left of the destination.
p.8
Conversion Between Number Systems
What is the process to convert a number from any base to decimal?
Multiply each digit by its place value and add the results.
p.8
Conversion Between Number Systems
How do you convert from decimal to another base?
The process involves dividing the decimal number by the base and recording the remainders.
p.9
Conversion Between Number Systems
What is the procedure for converting a decimal number to any base?
It requires two procedures: one for the integral part and one for the fractional part.
p.9
Conversion Between Number Systems
How is the integral part of a decimal number converted to another base?
By using repetitive division.
p.5
Hexadecimal System (Base 16)
How is the value of a hexadecimal number calculated?
Using the formula N = S_K * 16^(K-1) + S_(K-1) * 16^(K-2) + ... + S_1 * 16^1 + S_0 * 16^0.
p.3
Binary System (Base 2)
What symbols are used in the binary system?
Only two symbols: 0 and 1.
p.1
Positional Number Systems
What is the difference between positional and nonpositional number systems?
In positional systems, the position of a symbol determines its value, while in nonpositional systems, the value does not depend on the position.
p.18
Positional Number Systems
What determines the value of a symbol in a positional number system?
The position a symbol occupies in the number.
p.7
Conversion Between Number Systems
How is the decimal number 686 calculated from the octal number (1256)₈?
1×8³ + 2×8² + 5×8¹ + 6×8⁰ = 686.
p.19
Binary System (Base 2)
Why is the binary system called binary?
It is called binary because it uses two symbols (0 and 1), with a base of 2.
p.19
Octal System (Base 8)
Why is the octal system called octal?
It is called octal because it uses eight symbols (0-7), with a base of 8.
p.13
Conversion Between Number Systems
How can a decimal fraction be converted to binary when the denominator is a power of two?
By aligning the fractions according to their decimal equivalent values.
p.12
Conversion Between Number Systems
How do you convert the decimal number 178.6 to hexadecimal?
Multiply by 16 and record the integral parts, using only one digit to the right of the decimal point.
Do Roman numerals have a representation for zero?
No, they lack a zero digit in their system.
p.5
Hexadecimal System (Base 16)
Why is the hexadecimal system used in computing?
It provides a more convenient representation of binary numbers.
p.2
Positional Number Systems
How are positive and negative numbers stored in computers?
Differently, with specific representations for each.
p.2
Decimal System (Base 10)
What is the formula to calculate the value of an integer in the decimal system?
N = S K 2 1 3 10 K 2 1 + S K 2 2 3 10 K 2 2 + ... + S 0 3 10 0.
What symbols are used in the octal system?
{0, 1, 2, 3, 4, 5, 6, 7}.
p.19
Decimal System (Base 10)
Why is the decimal system called decimal?
It is called decimal because it is based on ten symbols (0-9), with a base of 10.
p.14
Conversion Between Number Systems
What is the relationship between binary and hexadecimal bases?
Four bits in binary is one digit in hexadecimal.
p.14
Conversion Between Number Systems
What is the relationship between binary and octal bases?
Three bits in binary is one octal digit.
p.9
Conversion Between Number Systems
How do you convert the binary number (110.11)₂ to decimal?
The conversion results in 6.75.
p.13
Positional Number Systems
How do you calculate the number of digits in a positional number system?
Using the formula K = ⎡ log_b N ⎤, where N is the decimal value.
p.11
Conversion Between Number Systems
How do you convert the decimal number 126 to hexadecimal?
By continuously dividing by 16 and recording the quotients and remainders, resulting in (7E)₁₆.
p.1
Introduction to Number Systems
What defines a number system?
A number system defines how a number can be represented using distinct symbols.
p.3
Binary System (Base 2)
What does the term 'bit' refer to in the binary system?
A binary digit, which can be either 0 or 1.
p.18
Conversion Between Number Systems
What is the process to convert a number from any system to decimal?
Multiply each digit by its place value and add the results.
p.18
Nonpositional Number Systems
What is a nonpositional number system?
A system where each symbol has a fixed value, regardless of its position.
p.6
Hexadecimal System (Base 16)
Is it common to represent real numbers in the hexadecimal system?
No, it is not very common.
p.19
Positional Number Systems
What is the difference between positional and nonpositional number systems?
In positional systems, the position of a digit affects its value; in nonpositional systems, the value is fixed regardless of position.
p.12
Conversion Between Number Systems
How do you convert the fractional part of a decimal number to binary?
Multiply the decimal number continuously by 2 and record the integral and fractional parts until the fractional part is 0 or enough bits are created.
p.12
Conversion Between Number Systems
How do you convert the decimal number 0.634 to octal?
Multiply by 8 and record the integral parts, stopping after four digits.
How is multiplication by 1000 expressed in Roman numerals?
By placing a bar above any of the six symbols (except I).
p.9
Conversion Between Number Systems
What is the decimal equivalent of the octal number (23.17)₈?
The decimal equivalent is approximately 19.234.
p.1
Decimal System (Base 10)
How are positive and negative numbers represented in the decimal system?
The sign (positive or negative) is indicated, but not stored in computers.
p.19
Positional Number Systems
What is the base or radix in a positional number system?
The base is the number of unique digits, including zero, used to represent numbers in that system.
p.19
Conversion Between Number Systems
Why is it easy to convert from binary to hexadecimal?
Because each hexadecimal digit corresponds to exactly four binary digits (bits).
p.14
Conversion Between Number Systems
What is the binary equivalent of the hexadecimal number (24C)?
Each hexadecimal digit converts to 4-bit patterns: 2 → 0010, 4 → 0100, C → 1100, resulting in (001001001100) in binary.
p.10
Conversion Between Number Systems
What is the first step in converting the integral part of a decimal number to another base?
Create an empty destination.
p.11
Conversion Between Number Systems
How do you convert the decimal number 126 to octal?
By continuously dividing by 8 and recording the quotients and remainders, resulting in (176)₈.
p.9
Conversion Between Number Systems
What is the decimal equivalent of the hexadecimal number (1A.23)₁₆?
The decimal equivalent is approximately 26.137.
p.10
Conversion Between Number Systems
What condition must be met to stop the conversion process?
When the quotient is zero.
p.5
Conversion Between Number Systems
What is the relationship between binary and decimal systems?
There is no obvious relationship between the number of bits in binary and the number of decimal digits.
p.1
Positional Number Systems
What is the significance of the base (radix) in a positional number system?
The base determines the total number of symbols in the set used to represent numbers.
p.2
Decimal System (Base 10)
What is the implicit aspect of writing numbers in the decimal system?
The plus sign is usually dropped but is understood.
p.19
Introduction to Number Systems
What is a number system?
A set of symbols and rules for representing and manipulating numbers.
p.14
Conversion Between Number Systems
How do you convert the binary number (10011100010) to hexadecimal?
Arrange it in 4-bit patterns: 100 1110 0010, which converts to (4E2) in hexadecimal.
p.15
Conversion Between Number Systems
What is the process to convert from octal to hexadecimal?
Convert the octal number to binary, then rearrange the bits in groups of four to find the hexadecimal equivalent.
p.10
Conversion Between Number Systems
How do you convert the decimal number 35 to binary?
By repeatedly dividing by 2 and recording the remainders.
p.4
Binary System (Base 2)
What is the formula to calculate the value of a binary integer?
N = S K * 2^(K-1) + S K-1 * 2^(K-2) + ... + S 1 * 2^1 + S 0 * 2^0.
p.1
Decimal System (Base 10)
What symbols are used in the decimal system?
The symbols used are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
p.4
Binary System (Base 2)
In the binary representation of a real number, how do the powers of K and L differ?
K starts from 0 and L starts from -1.
p.15
Conversion Between Number Systems
How do you convert the octal number (24)₈ to binary?
By writing each octal digit as its equivalent bit pattern, resulting in (010100)₂.
p.15
Conversion Between Number Systems
What is the process to convert from hexadecimal to octal?
Convert the hexadecimal number to binary, then rearrange the bits in groups of three to find the octal equivalent.
p.15
Maximum Values in Number Systems
How do you determine the minimum number of digits needed in the destination system?
If K digits are used in base b₁, the maximum number represented is b₁^K - 1.
p.16
Nonpositional Number Systems
What is the formula to find the minimum number of binary digits required to store decimal integers?
x = ⎡ K × (log b1 / log b2) ⎤
What is the restriction on the placement of symbols in Roman numerals?
A symbol S1 cannot come before another symbol S2 if S1 is greater than or equal to 10 times S2.
p.9
Conversion Between Number Systems
Why is the decimal result of (1A.23)₁₆ not exact?
It is rounded to three digits, as the exact value is 26.125.
p.5
Hexadecimal System (Base 16)
What symbols are used in the hexadecimal system?
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}
p.16
Nonpositional Number Systems
What is a nonpositional number system?
A system where the position of a symbol does not affect its value; each symbol has a fixed value.
p.5
Hexadecimal System (Base 16)
What do the symbols A, B, C, D, E, F represent in decimal?
10, 11, 12, 13, 14, and 15 respectively.
p.16
Nonpositional Number Systems
What is an example of a nonpositional number system?
The Roman numeral system.
p.3
Decimal System (Base 10)
How is a real number represented in the decimal system?
As R = S_K * 10^(K-1) + S_(K-1) * 10^(K-2) + ... + S_1 * 10^0 + S_0 * 10^(-1) + ... + S_L * 10^(-L).
p.4
Binary System (Base 2)
How can an integer be represented in the binary system?
As (S K –1 … S 1 S 0)₂, where S i is a digit.
p.5
Hexadecimal System (Base 16)
What is the significance of the place values in the hexadecimal system?
They represent powers of 16.
p.18
Introduction to Number Systems
What is a number system?
A system that uses distinct symbols to represent a number.
p.4
Binary System (Base 2)
What does the subscript '2' indicate in a binary number?
It indicates that the base is 2.
p.2
Decimal System (Base 10)
How is a number typically written in the decimal system?
As S K 2 1 … S 2 S 1 S 0 . S 2 1 S 2 2 … S 2 L, but often without parentheses, base, and plus sign.
p.3
Binary System (Base 2)
Why do computers use the binary system?
Because they are made of electronic switches that can only be in two states: on and off.
p.4
Binary System (Base 2)
How is a real number represented in the binary system?
As (S K –1 … S 1 S 0 · S –1 … S –L)₂, with K bits on the left and L bits on the right.
p.1
Decimal System (Base 10)
What does the term 'decimal' derive from?
The term 'decimal' is derived from the Latin root 'decem', which means ten.
p.4
Binary System (Base 2)
What is the calculation for the value of a real number in binary?
R = S K * 2^(K-1) + ... + S 0 * 2^0 + S -1 * 2^(-1) + ... + S -L * 2^(-L).
p.11
Conversion Between Number Systems
What is the process for converting the fractional part of a decimal number?
Multiply the fractional part by the base, insert the integral part into the destination, and repeat with the new fractional part.
p.1
Decimal System (Base 10)
What is the base of the decimal system?
The base of the decimal system is 10.
p.2
Decimal System (Base 10)
What is an integer in the decimal system?
An integral number with no fractional part.
p.2
Decimal System (Base 10)
What are place values in the decimal system?
Powers of 10 (10^0, 10^1, ..., 10^(K-1)).
p.2
Decimal System (Base 10)
What does the digit in position 1 represent in terms of value?
It has the value of 20, while the same digit in position 2 has the value of 200.
p.19
Hexadecimal System (Base 16)
Why is the hexadecimal system called hexadecimal?
It is called hexadecimal because it uses sixteen symbols (0-9 and A-F), with a base of 16.
p.1
Introduction to Number Systems
What is the role of symbols in number systems?
Symbols (digits) are used to represent numbers, similar to how characters create words in a language.
p.7
Comparison Between Number Systems
How is the number 15 represented in the four positional systems?
Decimal: 15, Binary: 1111, Octal: 17, Hexadecimal: F.
How is an integer represented in the octal system?
N = S_K * 8^K + S_(K-1) * 8^(K-1) + ... + S_1 * 8^1 + S_0 * 8^0.
