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Computer Arit.hmetic (Integers)

* Why several choices for representation

* The Choices 2' s complement 1 's complement Signed magnitude Long integers - l( . .... ~~i .. ~ .:-. .x

Decimal (BCD) Residue Arithmetic

• * Why several choices?

' '

Application space should drive architecture ·

- Compute intensive, low 1/0 - Arbitrarily. large precision - Generally within a fixed set,

e with opt.ion to go to multiples of that size ·

* The concept of "Long Integer" vs "Short ln!~ger" ·

Three Most Common Schemes

.. 2's complement 1 's complement Signed magnitude

Exam.pie: (A 4-bit data· pcith)

Representation What is being Represented

2's Comp 1 's Comp Sign-Mag

0000 0 0 0 0001 1 1 1 0010 ·2 2 2 0011 3 3 3 0100 4 4 4 0101 5 5 5 0110 6 6 6 0111 7 7 7 1000 -8 -7 0 1001 -7 -6 2 1010 -6 -5 2 1011 - 5 , . -4 3 1100 -4 -3 4 1101 -3 -2 5 1110 -2 - 1 6 1111 -1 -0 7

* Why 2's Complement?(Easy for Computer) * Why 1 's Complement (Self-Delusion) * Why Signed-Magnitude? (Easy for Humans)

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ResidueArithmetic

* When? - · Inputs, outputs are short

integers · - Intermediate results may be

very large - Internally compute intensive,

as opposed to having to do substantial 1/0

* How?

a,b

a* b

Transform to Residue Representation

(Slow)

I I Inverse transform

(Slow)

f(a), f(b) Perform the operation

(VERY FAST)

f(a) * f(b)

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