Homework 06 Solution

More Basic Conversion

In this problem we'll convert from base 10 to arbitrary bases and vice versa. However, because bases higher than 10 require that new "digits" be introduced, in this problem we'll stick to bases that are between 2 and 10 (unless you do the bonus part described at the very end of this page!)


Part 1

Write a Python function called numToBaseB(N, B) that takes as input a non-negative (0 or larger)

integer N and a base B (between 2 and 10 inclusive) and returns a string representing the number N in base B. (The rightmost digit of the string is the least significant digit of the number in base B.) Your code should output the string '0' when the input value of N is 0.

Remember, that your function is returning a string, and not a numeric value. Here are the Python functions for converting back and forth between strings and numbers:

str(x) returns the string representation of the number x. x may be a float or int.

int(s) returns the integer value of the string s. If s doesn't represent an int, Python stops with an error.
float(s) returns the floating-point value of the string s. If s doesn't represent a float, Python stops with an error.

Here are some sample runs:

    • numToBaseB(4, 2) '100'
    • numToBaseB(4, 3)

'11'

    • numToBaseB(4, 4)

'10'

    • numToBaseB(0, 4)

'0'

    • numToBaseB(0, 2)

'0'
CS 115 – Hw 6


Notice that the output of numToBaseB never has leading 0's. For example, when converting the base -10 number 4 to base 2, numToBaseB returned '100' and not '0100' nor '00000100'. Your implementation of numToBaseB should also not have leading 0's in its output.

Hint: Returning the empty string when the input value of N is 0 will help you to make sure other outputs

do not have leading zeros. You should do this inside a nested helper function, and it will become the base case of the recursive function. However, if the value of N is 0, the numToBaseB function must ultimately

return the string '0'.

Part 2

Naturally, we'd like to do the opposite conversion as well! Write a Python function

called baseBToNum(S, B) that takes as input a string S and a base B where S represents a number in base B where B is between 2 and 10 inclusive. The rightmost character of the string should be
the least significant digit of the number in base B—this is the familiar representation. baseBToNum should return an integer in base 10 representing the same number as S. Note that if the input string S is the empty string, the function baseBToNum should return 0.

Here are some sample runs:

    • baseBToNum("11", 2)

3

    • baseBToNum("11", 3)

4

    • baseBToNum("11", 10)

11

    • baseBToNum("", 10)

0    # the empty string should return 0

Part 3

Now, we can assemble what we've written to write a function called

called baseToBase(B1,B2,SinB1) that takes three inputs: a base B1, a base B2 (both of which are between 2 and 10, inclusive) and SinB1, which is a string representing a number in

base B1. baseToBase should return a string representing the same number in base B2. Here is some sample input and output:

    • baseToBase(2, 10, "11")# 11 in base 2 is 3 in base 10...

'3'

>>> baseToBase(10, 2, "3")    # 3 in base 10 is 11 in base 2...
CS 115 – Hw 6


'11'

>>> baseToBase(3, 5, "11")    # 11 in base 3 is 4 in base 5...

'4'

Again, the output of baseToBase should have no leading 0's. So, the output '11' in the second example above is OK; however, the output '011' would not be. Note that you don't have to rewrite

your numToBaseB function—instead, you can simply use numToBaseB and its twin, baseBToNum !

Part 4

Here's a short problem that puts what you've written to use! Write a program called add(S,T) that takes two binary strings S and T as input and returns their sum, also in binary. You can do this by converting the two binary strings to two base-10 numbers, adding the two numbers, and then converting the resulting sum back into base 2! Here is some sample input and output:

    • add("11", "01") '100'
    • add("011", "100") '111'
    • add("110", "011") '1001'
Part 5

add shows one way of adding two binary numbers: first convert them to base 10, add them, and then convert the result back to binary. In this problem, you will implement a different, more direct, method for adding two binary numbers:

101110

100111

--------

which, after the addition would look like this:

111

101110

100111

--------
CS 115 – Hw 6


1010101

Here the "carry" bits are in red.

For this problem, write a Python function called addB that takes two strings as input. These strings are

the representations of binary numbers. As usual, the rightmost bit of a string is the least significant bit. Your addB function should return a new string representing the sum of the two input strings. The sum
needs to be computed using the binary addition algorithm, shown above, and not using base conversions.

Your implementation should permit the input strings to have potentially different lengths and should output the sum without leading 0's. You may wish to define one or more "helper" functions that are used by addB (but you won't have to). Here is some sample input and output:

    • addB("11", "1") '100'
    • addB("011", "100") '111'

Note that addB should work entirely in binary, and entirely with strings. Do not use int() and str() to convert between the two data types.