What is the exact number of bytes in a system that contains
(a) 32K bytes, (b) 64M bytes, and (c) 6.4G bytes?
2) What is the largest binary number that can be expressed with 16 bits?
What are the equivalent decimal and hexadecimal numbers?
The Correct Answer and Explanation is:
1. Number of Bytes in a System:
To determine the number of bytes in the given system sizes, let’s first clarify the terms:
- K stands for kilobyte, where 1 K = 1024 bytes.
- M stands for megabyte, where 1 M = 1024 KB = 1024 × 1024 bytes.
- G stands for gigabyte, where 1 G = 1024 MB = 1024 × 1024 × 1024 bytes.
Now, let’s calculate the number of bytes for each:
(a) 32K bytes:
- ( 32K = 32 \times 1024 \, \text{bytes} = 32768 \, \text{bytes} )
Thus, a system containing 32K bytes has 32,768 bytes.
(b) 64M bytes:
- ( 64M = 64 \times 1024 \times 1024 \, \text{bytes} = 67108864 \, \text{bytes} )
Thus, a system containing 64M bytes has 67,108,864 bytes.
(c) 6.4G bytes:
- ( 6.4G = 6.4 \times 1024 \times 1024 \times 1024 \, \text{bytes} = 6871947673.6 \, \text{bytes} )
- Rounded, this is approximately 6,871,947,674 bytes.
2. Largest Binary Number with 16 Bits:
In a system that uses 16 bits, the largest binary number can be expressed by setting all 16 bits to 1. This would be:
- Binary: ( 1111111111111111_2 )
To Convert to Decimal:
The value of this binary number is the sum of powers of 2 for each position where there is a 1. Since all 16 bits are set to 1, we calculate the value as:
- Decimal: ( 2^{15} + 2^{14} + \dots + 2^1 + 2^0 )
- This equals ( 65535 ) in decimal.
Thus, the largest binary number that can be expressed with 16 bits is:
- Binary: ( 1111111111111111_2 )
- Decimal: 65535
- Hexadecimal: ( FFFF_{16} )
Explanation:
The number of bytes in a system depends on the prefixes (K, M, G) and how they relate to powers of 2. The base-2 system used in computer memory means that we use multiples of 1024 to calculate the size in bytes, rather than powers of 10 as is commonly used in everyday measurements.
For the largest binary number expressed in 16 bits, we observe that the binary system uses two possible values (0 or 1) per bit. With 16 bits, the highest possible combination is all bits being 1, which gives the largest binary value for that bit length. This binary value can be converted to decimal and hexadecimal by understanding how each place value represents a power of 2, and converting it directly from binary to the respective numeral system. The result is the largest possible value that can be represented by 16 bits.