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CSE 20: Discrete Mathematics for Computer Science Prof. Shachar Lovett 2 Todayβs Topics: 1. 2. Number representations in different bases Converting between bases 3 1. Number representations all your base are belong to us Numbers are building blocks 5 Bases ο Base πβ₯2 ο Number ο ο ο In in base b: dkdk-1β¦d1d0 Digits between 0 and b-1 Number is d0+d1*b+d2*b2+β¦+dk*bk every base b, any integer has a unique representation in this base Values in different bases Whatβs the decimal value of (10001)2? A. B. C. D. E. (5)10 (17)10 (-1)10 (10001)10 None of the above / more than one of the above. Values in different bases Whatβs the base 2 representation of the decimal number (42)10? A. B. C. D. E. (111111)2 (100001)2 (101010)2 (110011)2 None of the above / more than one of the above. Values in different bases Whatβs the biggest integer value whose binary representation has 4 bits? A. B. C. D. E. 24 = (16)10 23 = (8)10 (4)10 (1000)10 None of the above / more than one of the above. Uniqueness Is it possible to have two different representations for an integer in base 2? That is, is it possible to have A. B. C. D. E. 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 ο Theorem: For any integer π β₯ 0 and any base π β₯ 2, there is exactly one way to write n in base b ο Need ο We to prove existence and uniqueness will prove for b=2; the proof can be extended to any b 11 Proof of existence (b=2) ο ο ο ο ο ο ο 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 β¦ d0)2 Then n=(dm β¦ 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) ο Case 2: n is odd, n=2k+1 ο Assume k=(dm β¦ d0)2 ο Then n=(dm β¦ d0 1)2 ο Why? ο We already proved that (dm β¦ d0 0)2=2k ο add 1 to both sides 13 Proof of uniqueness (b=2) ο Assume n=(dmβ¦d0)2 = (etβ¦e0)2 ο Want to prove: must be the same ο Need to assume that the most significant digits are nonzero, which in binary means they are 1 ο So assume dm=et=1 ο We will prove m=t and di=ei for all i 14 Proof of uniqueness (b=2) ο Proof by strong induction ο Base: n=0, only way is (0)2 ο Inductive: assume for all k<n, prove for n ο We know least significant digit is n MOD 2, so d0=e0=(n MOD 2) ο We βpeelβ the least digit: n DIV 2 = (dmβ¦d1)2 = (etβ¦e1)2 ο By strong induction, must be the same Parity and shift Shifts Values in different bases Whatβs the base 2 representation of the decimal number (2014)10 A. B. C. D. E. (11111011110)2 (10000000000)2 (10101010101)2 (1000000001)2 None of the above / more than one of the above. Values in different bases Whatβs the base 2 representation of the decimal number (2014)10 A. B. C. D. E. (11111011110)2 (10000000000)2 (10101010101)2 (1000000001)2 None of the above / more than one of the above. Is there a systematic way (aka algorithm) to do it? Decimal to Binary conversion toBinary(pos int n) Begin binary=ββ i=n While i>0 Do If (i is even) Then binary=β0β+binary End If (i is odd) Then binary=β1β+binary End i=i DIV 2 Output binary End. ο Right to left ο Questions to ask: ο Does it always terminate? ο Does it give the correct answer? ο What is the time complexity? Other numbers? ο Fractional components ο Negative numbers aka how to subtract A. B. C. β¦ first, how do we add? D. E. 111 100 1011 1111 Other One bit addition Carry: 1 101 +110 1011 Subtraction 6 ο Borrowing A β B = (A β 10) + (10 β B) ο Carrying A β B = (A+10) β (B+10) ο Complementation A β B = A + Bc = A + [ (99-B) - 99 ] = A + [ (100-B) β 100 ] JS p. 2βs complement 0000 1111 1110 1101 1100 -2 0 0010 1 2 -3 3 4 -4 -5 -6 Complete the wheel of numbers! -1 0001 5 -7 6 -8 1001 1000 How many numbers are we representing with 4 bits? 7 0111 0011 How to add binary numbers? 1 1 0 0 1 0 1 0 1 +1 0 1 1 0 1 1 0 1 ? ? ? ? ? ? ? ? ? ? How to add binary numbers? ? ? 1 +1 ? ? ? 1 0 ? ? 0 1 ? ? 0 1 ? ? 1 0 ? ? 0 1 ? ? 1 1 ? ? carry 0 1 0 1 ? ? How to add binary numbers? ? ? 1 +1 ? ? ο Two ? 1 0 ? ? 0 1 ? ? 0 1 ? ? 1 0 ? ? 0 1 ? basic operations: ο One-Bit-Addition(bit1, bit2, carry) ο Next-carry(bit1, bit2, carry) ? 1 1 ? ? carry 0 1 0 1 ? ? Numbers β¦ logic β¦ circuits