Lecture 2: Logic Gates & Fibonacci Numbers¶

Date: 09/05/2017, Tuesday

In [1]:

format compact

Logic gates¶

nand gate¶

In [2]:

nand = @(a,b) ~(a&b)

nand =
  function_handle with value:
    @(a,b)~(a&b)

In [3]:

nand(1,1) % test if it works

ans =
  logical
   0

print truth table¶

In [4]:

help boolean_print_TT_fn % boolean_print_TT_fn.m is available on canvas

  boolean_print_TT_fn.m
  a function to print the boolean truth table for a given
  supplied function 'func'

  INPUT
  func: the supplied function (e.g. OR, NAND, XOR)
  input_num: number of inputs

In [5]:

boolean_print_TT_fn(nand,2)

_______
a|b|out
0|0| 1
0|1| 1
1|0| 1
1|1| 0

build “not” gate from “nand” gate¶

In [6]:

my_not = @(a) nand(a,a) % "not" is a built-in function so we use my_not to avoid conflicts

my_not =
  function_handle with value:
    @(a)nand(a,a)

In [7]:

boolean_print_TT_fn(my_not,1)

_____
a|out
0|1
1|0

Why nand(a,a) means not:

and(a,a) = a, no matter a is 0 or 1
nand(a,a) = not and(a,a) = not a

Fibonacci Sequence¶

generate Fibonacci sequences¶

In [8]:

%%file fib_fn.m
function F = fib_fn(n)

    F = zeros(1,n);
    F(1)=1;
    F(2)=1;
    for j=3:n
        F(j)=F(j-1)+F(j-2);
    end

end

Created file '/Users/zhuangjw/Research/Computing/personal_web/AM111/docs/fib_fn.m'.

In [9]:

fib_fn(10)

ans =
     1     1     2     3     5     8    13    21    34    55

Compare with the built-in function fibonacci( )

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fibonacci(1:10)

ans =
     1     1     2     3     5     8    13    21    34    55

golden ratio¶

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golden_ratio = (sqrt(5)+1)/2 % true value

golden_ratio =
    1.6180

\(F_n/F_{n-1}\) Converges to the golden ratio

In [12]:

F = fib_fn(10);
F_ratio = F(2:end)./F(1:end-1)

F_ratio =
  Columns 1 through 7
    1.0000    2.0000    1.5000    1.6667    1.6000    1.6250    1.6154
  Columns 8 through 9
    1.6190    1.6176

In [13]:

%plot --size 400,300
hold on
plot(F_ratio,'-o')
plot([0,10],[golden_ratio,golden_ratio])