Adding Numbers

Adding in Base 10

We know how to add in base 10. What is the algorithm?

Adding in Base 10

Assume numbers a and b have n digits

 

carry = 0

d = ""

for i = n - 1 downto 0  # Right to left

    sum = a[i] + b[i] + carry

    d[i] = sum mod 10

    carry = sum / 10

# We also have the carry here

Adding in Base 2

Now that we have an algorithm for adding in base 10, what would the algorithm be for adding in base 2?

Adding in Base 2

Assume numbers a and b have n digits

 

carry = 0

d = ""

for i = n - 1 downto 1  # Right to left

    sum = a[i] + b[i] + carry

    d[i] = sum mod 2

    carry = sum / 2

# We also have the carry here

Integer Overflow

Signed 32-bit Integers

Maximum size of a signed 32-bit integer is about 2 billion. What happens when we exceed this?

 

Things that are approaching or exceeded this limit:

* Gangnam Style

* Twitter post ids (Unsigned)

* IPv4 (Unsigned)

* Unix Time

64-bit and 128-bit numbers

How large are these numbers?

Multiplication

Algorithm for Multiplication

This algorithm is kinda gnarly. What would this look like at a higher level?

Base 2 Multiplication

Multiplication in base 2 is actually easy. We need the concept of a left shift, where we add n zeros to the right of the number. We're shifting the number n places. You can also think of it as multiplying by the base.

 

There is also a right shift, where we bring zeros or ones in from the left.

 

What would an algorithm look like?

Base 2 Multiplication

products = []

for i = n - 1 downto 0 # Right to left again.

  if b[j] == 1

      products[j] = a[j] << (i - n + 1)

  else

      products[j] = 0

 

# sum all the values in products.

Floating Point

Representing fractions

Computers are only able to handle 1's and 0's. How do we do fractional numbers?

 

We represent them in "scientific notation".

M x 10e where M = Mantissa and 0 < M < 1 and e = exponent.

Representation in memory

A floating point number contains 3 parts.

 

1 bit for the sign

8 or 11 bits for the exponent

23 or 52 bits for the mantissa

 

Modulo Arithmetic

Division and Remainders

When we typically talk about division with integers, we care about the quotient and the remainder.

 

The quotient of a divided by b is denoted by "a div b".

The remainder of a divided by b is denoted by "a mod b".

Modulo Congruences

Two numbers are congruent modulo m if
a mod m = b mod m. That is, for an integer k, we have:

 

a = b + km

 

And we say a ≡ b (mod m).

 

Example:

8 mod 12 = 20 mod 12.

Modulo Congurences

Theorem 1:

 

If a ≡ b (mod m) and c ≡ d (mod m), then:

 

a + c ≡ b + d (mod m)

a * c ≡ b * d (mod m)

Modulo Congurences

Corollary:

 

m is a positive integer and a and b be integers. Then:

 

(a + b) mod m = ((a mod m) + (b mod m)) mod m

(a * b) mod m = ((a mod m) * (b mod m)) mod m

Gotchas

If ac ≡ bc mod m, you may think that a ≡ b, but this is not necessarily true.

 

Also keep in mind the previous slide.

Arithmetic Modulo m

We can define arithmetic on the set of nonnegative integers less than m.

 

a +_m b = (a + b) mod m

a *_m b = (a * b) mod m

 

Similar to adding time to a clock.

Why???

Hashing Functions

You may have heard of sets, hashes, or dictionaries. Internally, these data structures all use an array.

 

They determine where to insert data based on hashing functions and modulo arithmetic.

Checksums

https://en.wikipedia.org/wiki/Cyclic_redundancy_check

Cryptography

https://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange#Cryptographic_explanation

https://en.wikipedia.org/wiki/RSA_(cryptosystem)#Operation