How Numbers Are Stored in Bytes

Topic: How numbers are stored in bytes.

• Video on overflow
• Video on negative numbers
• Video on endianess
• Negative numbers another perspective

Questions

• Explain how to derive the maximum number that can be stored in one byte. What about two bytes?
• Why is this maximum number one less than the number of possible combinations of bits in those positions?
• Given a number x (for example, 0000 0011), what must be done to obtain the representation of its negative (in the two’s complement system)? Derive a formula/algorithm (1111 1101).
• Under what conditions does $a + b < a$ if both $a$ and $b$ are positive and represented using 8 bits?
• What digits are used in the hexadecimal system?
• How do you convert 0101 1111 to hexadecimal?
• In which other numeral systems can an 8-bit value be written without performing calculations?

  hint

  A group of X bits corresponds to one digit in that system.

  hint 2

  In the hexadecimal numeral system, X = 4.

  hint 3

  Grouping the bits in pairs gives 01 01 11 11. Each of these values can be represented by a single digit in a numeral system where the number of digits equals the number of values that can be represented by 2 bits. Two bits can represent 4 values (00, 01, 10, 11). That is, we can use the base-4 system: 01 01 11 11 -> 1 1 3 3 -> 1133.

  question

  Why must the system’s base be a power of two?