1. CSE 20: Discrete Mathematics for Computer Science Prof. Shachar Lovett

2. Today’s Topics: Number representations in different bases
Converting between bases

3. 1. Number representations
all your base are belong to us

4. Numbers are building blocks
Five

5. Bases Base 𝑏≥2 Number in base b: dkdk-1…d1d0
Digits between 0 and b-1
Number is d0+d1*b+d2*b2+…+dk*bk
In every base b, any integer has a unique representation in this base

6. Values in different bases
What’s the decimal value of
(10001)2?
(5)10
(17)10
(-1)10
(10001)10
None of the above / more than one of the above.

7. Values in different bases
What’s the base 2 representation of the decimal number
(42)10?
(111111)2
(100001)2
(101010)2
(110011)2
None of the above / more than one of the above.

8. Values in different bases
What’s the biggest integer value whose binary representation has 4 bits?
24 = (16)10
23 = (8)10
(4)10
(1000)10
None of the above / more than one of the above.

9. Uniqueness
Is it possible to have two different representations for an integer in base 2? That is, is it possible to have
No.
Yes, but m has to be the same as n.
Yes, and m,n can be different but for each kind of coefficient that appears in both, it has to agree. That is, a0 = b0, a1 = b1, etc.
Yes, if m=n and all the coefficients agree.
More than one of the above / none of the above.

10. Existence and uniqueness
Theorem: For any integer 𝑛≥0 and any base 𝑏≥2, there is exactly one way to write n in base b
Need to prove existence and uniqueness
We will prove for b=2; the proof can be extended to any b

11. Proof of existence (b=2)
Proof by strong induction:
Base: n=0 = (0)2
Inductive: assume for all k<n, prove for n
Case 1: n is even, n=2k
Assume k=(dm … d0)2
Then n=(dm … d0 0)2
Why? Adding a 2 to the right multiplies the number by 2 (shifts all digits by 1)

12. Proof of existence (b=2)
Case 2: n is odd, n=2k+1
Assume k=(dm … d0)2
Then n=(dm … d0 1)2
Why?
We already proved that (dm … d0 0)2=2k
add 1 to both sides

13. Proof of uniqueness (b=2)
Assume n=(dm…d0)2 = (et…e0)2
Want to prove: must be the same
Need to assume that the most significant digits are nonzero, which in binary means they are 1
So assume dm=et=1
We will prove m=t and di=ei for all i

14. Proof of uniqueness (b=2)
Proof by strong induction
Base: n=0, only way is (0)2
Inductive: assume for all k<n, prove for n
We know least significant digit is n MOD 2, so d0=e0=(n MOD 2)
We “peel” the least digit:
n DIV 2 = (dm…d1)2 = (et…e1)2
By strong induction, must be the same

15. Parity and shift

16. Shifts

17. Values in different bases
What’s the base 2 representation of the decimal number
(2014)10
( )2
( )2
( )2
( )2
None of the above / more than one of the above.

18. Values in different bases
What’s the base 2 representation of the decimal number
(2014)10
( )2
( )2
( )2
( )2
None of the above / more than one of the above.
Is there a systematic way (aka algorithm) to do it?

19. Decimal to Binary conversion
toBinary(pos int n)
Begin
binary=“”
i=n
While i>0 Do
If (i is even) Then
binary=“0”+binary
End
If (i is odd) Then
binary=“1”+binary
i=i DIV 2
Output binary
End.
Right to left
Questions to ask:
Does it always terminate?
Does it give the correct answer?
What is the time complexity?
Biggest power of 2 less than.
Recursive. Connect back to RPM.

20. Other numbers? Fractional components
Negative numbers aka how to subtract
… first, how do we add?
111
100
1011
1111
Other

21. One bit addition 1
Carry:
1 0 1

22. Subtraction JS p. 6 Borrowing Carrying Complementation
A – B = (A – 10) + (10 – B)
Carrying
A – B = (A+10) – (B+10)
Complementation
A – B = A + Bc = A + [ (99-B) - 99 ] = A + [ (100-B) – 100 ]

23. 2’s complement
How many numbers are we representing with 4 bits?
0000
1111
0001
1110 -1 1
0010 -2 2
1101
0011 -3 3
1100 -4 4 5 -5 -6 6 -7
Complete the wheel of numbers! 7 -8
1001
0111
1000

24. How to add binary numbers?
? ? ? ? ? ? ? ? ? ?

25. How to add binary numbers?
? ? ? ? ? ? ? ? ? carry
? ? ? ? ? ? ? ? ? ?

26. How to add binary numbers?
? ? ? ? ? ? ? ? ? carry
? ? ? ? ? ? ? ? ? ?
Two basic operations:
One-Bit-Addition(bit1, bit2, carry)
Next-carry(bit1, bit2, carry)

27. Numbers … logic … circuits