Integer Type Representation

Question 1

Integer Type
Representation

Answer 1

• The standard makes few guarantees
  about how these types should be
  represented.
• It does for unsigned types.

Question 2

Unsigned Values in Binary

Answer 2

• For humans (in base ten)
  – Each digit can have a value of 0 .. 9
  – Each digit is worth a power of 10.
• Consider a (base ten) number like 3725
  310^3, 710^2, 210^1, 510^0
• This is called a positional number system
• Computers are really good at storing on / off
• Binary numbers use this store arbitrary values
  – We just have 2 digits, 0 (i.e., off) and 1 (on)
  – Place value uses powers of 2 rather than powersof ten.
• Consider a binary number like 1010111
  12^6, 12^5, 12^4, 12^3, 12^2, 12^1, 1*2^0

Question 3

Decimal to Binary

Answer 3

• Given a number in base 10
• How can we figure out it value in binary?
• We’ll see two common ways to do this.
• Each bit contributes to a sum.
  – Each one can be either 0 or 1.
  – If it’s zero, it contributes nothing to the sum.
  – If it’s one, it contributes a power of 2.
• Try a 1 for the most significant bit.
  – The bit on the left.
• Here, if the 26 bit was 1, the sum would be too large.
• Try out a 1 for the 25 bit.
• That adds 32 to the sum.
  – That’s less than (or equal to) 51.
  – … and, all the remaining bits only add up to 31.
  – … so the 25 bit must be a 1
• That accounts for 32 of the 51 we started with
  – Leaving 19 (of the original 51)
  – … the remaining bits must add up to 19
• The 24 bit is worth 16
  – That doesn’t exceed the 19 we have remaining.
  – … and all remaining bits add up to just 15.
  – … so the 24 bit must be a 1.
• That leaves 3 for the rest of the bits.
• The 23 bit must be zero
  – Otherwise the sum would be too large
• Likewise, the 22 bit must be zero
• The 21 bit must be a 1.
  – It’s worth 21 = 2
• That leaves just 1 for the last bit
• The least significant bit must be a 1.
  – The one on the right.
• The value of all the bits add up to 51.
• This technique is called reduction of powers
• How do you know what bit to start with?
  – Two answers
  – You can try out larger and larger powers of two
• Until you find one that’s larger than the value you’re
  converting.
  – But, usually, you’re working with a fixed number
  of bits, based on the type.
• E.g., an 8-bit char, a 16-bit short or a 32-bit it.