E2011: Theoretical fundamentals of computer science
Topic 3: Numeral systems - Exercises
Vlad Popovici, Ph.D.
Fac. of Science - RECETOX
Problem 1
Implement a 2-bit adder using logical gates.
Vlad Popovici, Ph.D. (Fac. of Science - RECETOX)E2011: Theoretical fundamentals of computer science Topic 3: Numeral systems - Exercise2 / 8
Plan
which numbers can be represented on 2 bits?
Vlad Popovici, Ph.D. (Fac. of Science - RECETOX)E2011: Theoretical fundamentals of computer science Topic 3: Numeral systems - Exercise3 / 8
Plan
which numbers can be represented on 2 bits?
what is the range of results?
Vlad Popovici, Ph.D. (Fac. of Science - RECETOX)E2011: Theoretical fundamentals of computer science Topic 3: Numeral systems - Exercise3 / 8
Plan
which numbers can be represented on 2 bits?
what is the range of results?
how many bits you need for the result?
Vlad Popovici, Ph.D. (Fac. of Science - RECETOX)E2011: Theoretical fundamentals of computer science Topic 3: Numeral systems - Exercise3 / 8
Plan
which numbers can be represented on 2 bits?
what is the range of results?
how many bits you need for the result?
write the truth table and derive the fuctions for the outputs
Vlad Popovici, Ph.D. (Fac. of Science - RECETOX)E2011: Theoretical fundamentals of computer science Topic 3: Numeral systems - Exercise3 / 8
Plan
which numbers can be represented on 2 bits?
what is the range of results?
how many bits you need for the result?
write the truth table and derive the fuctions for the outputs
design the circuit
Vlad Popovici, Ph.D. (Fac. of Science - RECETOX)E2011: Theoretical fundamentals of computer science Topic 3: Numeral systems - Exercise3 / 8
Solution
Input: a = [a1a0], b = [b1b0].
Output: s = [cs1s0]; c: carry
Truth table:
a1 a0 b1 b0 c s1 s0
0 0 0 0
0 0 0 1
0 0 1 0
0 0 1 1
0 1 0 0
0 1 0 1
0 1 1 0
0 1 1 1
1 0 0 0
1 0 0 1
1 0 1 0
1 0 1 1
1 1 0 0
1 1 0 1
1 1 1 0
1 1 1 1
Vlad Popovici, Ph.D. (Fac. of Science - RECETOX)E2011: Theoretical fundamentals of computer science Topic 3: Numeral systems - Exercise4 / 8
Solution
Input: a = [a1a0], b = [b1b0].
Output: s = [cs1s0]; c: carry
Truth table:
a1 a0 b1 b0 c s1 s0
0 0 0 0 0 0 0
0 0 0 1 0 0 1
0 0 1 0 0 1 0
0 0 1 1 0 1 1
0 1 0 0 0 0 1
0 1 0 1 0 1 0
0 1 1 0 0 1 1
0 1 1 1 1 0 0
1 0 0 0 0 1 0
1 0 0 1 0 1 1
1 0 1 0 1 0 0
1 0 1 1 1 0 1
1 1 0 0 0 1 1
1 1 0 1 1 0 0
1 1 1 0 1 0 1
1 1 1 1 1 1 0
Vlad Popovici, Ph.D. (Fac. of Science - RECETOX)E2011: Theoretical fundamentals of computer science Topic 3: Numeral systems - Exercise5 / 8
Vlad Popovici, Ph.D. (Fac. of Science - RECETOX)E2011: Theoretical fundamentals of computer science Topic 3: Numeral systems - Exercise6 / 8
Problem 2
Using bitwise operations, extract the R, G, B values from a HTML-like
specification (in hexa) of the form ”#RRGGBB”, where each symbol
corresponds to a hexa digit. Example, from ”#ABCDEF”, you should get
R=”AB”, G=”CD”, B=”EF”.
Vlad Popovici, Ph.D. (Fac. of Science - RECETOX)E2011: Theoretical fundamentals of computer science Topic 3: Numeral systems - Exercise7 / 8
Solution
RGB
R
ed
(R
)rrrr rrrr
23 16
G
reen
(G
)
gggg gggg
15 8
B
lue
(B
)
bbbb bbbb
7 0
let x be the input value (on 24 bits, i.e. 6 bytes)
R = x >> 16 (right shift by 16 bits)
G = (x << 4) >> 16 (left shift followed by right shift)
B = x&FF (bitwise AND)
can you see what happened in each case?
can you find other solutions?
Vlad Popovici, Ph.D. (Fac. of Science - RECETOX)E2011: Theoretical fundamentals of computer science Topic 3: Numeral systems - Exercise8 / 8