What is the base of the octal number system?
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What is the base of the octal number system?
What do we call the integral part of the decimal number in the conversion process?
Source.
What do we call the integral part of the converted number?
Destination.
What is the octal result of converting the decimal number 0.634?
(0.5044)₈.
What is the largest six-digit decimal number?
999,999.
What is the hexadecimal equivalent of the decimal number 126?
(7E)₁₆.
What happens when a smaller symbol is placed before a larger symbol in Roman numerals?
The smaller value is subtracted from the larger one.
What is done with the remainder during the conversion process?
The remainder is inserted to the left of the destination.
What is the binary representation of the decimal number 5.75?
(101.11)₂
What is the process to convert a number from any base to decimal?
Multiply each digit by its place value and add the results.
How do you convert from decimal to another base?
The process involves dividing the decimal number by the base and recording the remainders.
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.
How is the integral part of a decimal number converted to another base?
By using repetitive division.
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.
What symbols are used in the binary system?
Only two symbols: 0 and 1.
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.
What determines the value of a symbol in a positional number system?
The position a symbol occupies in the number.
What is the decimal equivalent of the hexadecimal number (2AE)16?
What is the equivalent decimal value of the binary number (11001)₂?
How is the decimal number 686 calculated from the octal number (1256)₈?
1×8³ + 2×8² + 5×8¹ + 6×8⁰ = 686.
Why is the binary system called binary?
It is called binary because it uses two symbols (0 and 1), with a base of 2.
Why is the octal system called octal?
It is called octal because it uses eight symbols (0-7), with a base of 8.
What is the octal equivalent of the binary number (101110010)₂?
(562)₈.
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.
What is the maximum number of decimal digits discussed in the example?
Six digits.
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.
Why is the hexadecimal system used in computing?
It provides a more convenient representation of binary numbers.
What do we call the fractional part of the converted number?
Destination.
What is the value of the symbol 'X' in the Roman numeral system?
How are positive and negative numbers stored in computers?
Differently, with specific representations for each.
What is the base of the decimal system?
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}.
How are four bits in the binary system represented in the hexadecimal system?
As one digit.
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.
What is the relationship between binary and hexadecimal bases?
Four bits in binary is one digit in hexadecimal.
What is the relationship between binary and octal bases?
Three bits in binary is one octal digit.
What is the rule for adding values in Roman numerals when a smaller symbol is placed after a larger or equal symbol?
The values are added.
How do you convert the binary number (110.11)₂ to decimal?
The conversion results in 6.75.
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.
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.
What is the number of digits required for the decimal number 234 in hexadecimal?
2
What is the number of digits required for the decimal number 234 in octal?
3
What is the hexadecimal representation of the decimal number 14?
E
What symbols are used in the Roman numeral system?
I, V, X, L, C, D, M.
What defines a number system?
A number system defines how a number can be represented using distinct symbols.
What does the term 'bit' refer to in the binary system?
A binary digit, which can be either 0 or 1.
What is the place value of the digit '4' in the number 24.13?
10^1 (or 10).
What is the formula for the maximum value of a hexadecimal integer with K digits?
N_max = 16^K - 1.
What is the octal representation of the decimal number 686?
(1256)₈.
What is the process to convert a number from any system to decimal?
Multiply each digit by its place value and add the results.
What is the Latin root of the word 'octal'?
Octo, meaning eight.
What is a nonpositional number system?
A system where each symbol has a fixed value, regardless of its position.
Is it common to represent real numbers in the hexadecimal system?
No, it is not very common.
What is the base of the binary system?
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.
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.
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).
What is the decimal equivalent of the octal number (23.17)₈?
The decimal equivalent is approximately 19.234.
What is the value of the Roman numeral 'III'?
What is the value of the Roman numeral 'IV'?
What is the value of the Roman numeral 'MMVII'?
What is the value of the symbol 'C' in the Roman numeral system?
How are positive and negative numbers represented in the decimal system?
The sign (positive or negative) is indicated, but not stored in computers.
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.
Why is it easy to convert from binary to hexadecimal?
Because each hexadecimal digit corresponds to exactly four binary digits (bits).
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.
What is the first step in converting the integral part of a decimal number to another base?
Create an empty destination.
What is the binary representation of the decimal number 165?
10100101
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.
What is the decimal equivalent of the hexadecimal number (1A.23)₁₆?
The decimal equivalent is approximately 26.137.
In the conversion algorithm, what becomes the new source after each division?
The quotient.
What is the number of digits required for the decimal number 234 in binary?
8
What is the binary representation of the decimal number 35?
What condition must be met to stop the conversion process?
When the quotient is zero.
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.
What is the base of the octal system?
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.
What is the implicit aspect of writing numbers in the decimal system?
The plus sign is usually dropped but is understood.
What is the shortest representation for the number 15 among the four systems?
Hexadecimal: F.
What is a number system?
A set of symbols and rules for representing and manipulating numbers.
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.
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.
What is the binary representation of the decimal number 5?
101
What is the octal representation of the decimal number 12?
14
How do you convert the decimal number 35 to binary?
By repeatedly dividing by 2 and recording the remainders.
What is the binary representation of the decimal number 15?
1111
What is the octal representation of the decimal number 8?
10
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.
What is the decimal equivalent of the binary number (101.11)₂?
5.75
What symbols are used in the decimal system?
The symbols used are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
What is the place value of the digit '3' in the number 24.13?
10^(-1) (or 0.1).
How many symbols are used in the hexadecimal system?
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.
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.
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.
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.
What is the binary result of converting the decimal number 0.625?
(0.101)₂.
How is the decimal number 10 represented in hexadecimal?
A
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 binary representation of the decimal fraction 27/64?
0.011011
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.
What is the octal equivalent of the decimal number 126?
(176)₈.
Why is the decimal result of (1A.23)₁₆ not exact?
It is rounded to three digits, as the exact value is 26.125.
What is the base of the hexadecimal system?
16
What is the hexadecimal result of converting the decimal number 178.6?
(B2.9)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}
What is the largest 20-bit binary number?
1,048,575.
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.
What do the symbols A, B, C, D, E, F represent in decimal?
10, 11, 12, 13, 14, and 15 respectively.
What is an example of a nonpositional number system?
The Roman numeral system.
What do we call the fractional part of the decimal number during conversion?
Source.
What is the maximum value of a decimal integer that can be represented by K digits?
N_max = 10^K - 1.
What is the value of the Roman numeral 'XIX'?
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).
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.
What is the significance of the place values in the hexadecimal system?
They represent powers of 16.
What is a number system?
A system that uses distinct symbols to represent a number.
What does the subscript '2' indicate in a binary number?
It indicates that the base is 2.
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.
Why do computers use the binary system?
Because they are made of electronic switches that can only be in two states: on and off.
What is the maximum value of a binary integer with K digits?
N max = 2^K - 1.
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.
What are the symbols in the binary system called?
Binary digits or bits.
What does the term 'decimal' derive from?
The term 'decimal' is derived from the Latin root 'decem', which means ten.
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).
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.
What is the number of digits required for the decimal number 234 in decimal?
3
What is the decimal representation of the binary number 111?
7
What is the base of the binary system?
The base is 2.
What is the value of the Roman numeral 'MDC'?
What is the base of the decimal system?
The base of the decimal system is 10.
What is an integer in the decimal system?
An integral number with no fractional part.
What are place values in the decimal system?
Powers of 10 (10^0, 10^1, ..., 10^(K-1)).
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.
What is the maximum value of an octal integer with K digits?
N_max = 8^K - 1.
What is the maximum value of a 5-digit octal number?
What are the symbols used in the octal system?
0, 1, 2, 3, 4, 5, 6, 7.
What is the base of the hexadecimal system?
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.
What is the maximum value of a hexadecimal integer with 5 digits?
1,048,575.
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.
How is the number 15 represented in the four positional systems?
Decimal: 15, Binary: 1111, Octal: 17, Hexadecimal: F.
How many bits in the binary system are represented by one digit in the octal system?
Three bits.
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.
Which positional number system uses base 10?
Decimal system.
How many bits in the binary system are represented by one digit in the hexadecimal system?
Four bits.
How are three bits in the binary system represented in the octal system?
As one digit.
