This week we will implement the basic arithmetic operations addition, subtraction, division and multiplication. Typically, these operations are implemented in the electronic circuitry of the CPU itself. In this weekly assignment, we will implement functions for these arithmetic operations with only making little use of the existing operators. For the implementation, we will refer to the algorithms that we were thought in high school and implement them in Python. When we have implemented these functions, we can start to solve simple arithmetic problems using our calculator.
- Loops
- Arithmetic
- Functions
- Conditions
You will implement what we will call the "high school addition algorithm". This algorithm will take two numbers, both represented in the decimal number system. As an example, take the numbers 234 and 456. Let these numbers consist of digits such that 123 consists of the digits 1, 2 and 3 and the number 456 consists of the digits 4, 5 and 6. In the decimal number system, digits can only take the values 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9.
This means that if we want to represent a number beyond the digit 9, we need one
extra digit. Let us consider the expression 9+3, the result of which will
overflow one digit:
| Carry | 1 | ||
|---|---|---|---|
| 9 | |||
| 3 | |||
| + | |||
| Result | 1 | 2 |
Adding the digits from right-to-left, we get:
Step 1: (Add Digits, start with right-most digits)
9 + 3 = 12
Since the number that we calculated is greater than 9, we need one more digit to represent this number. We say that, since the result of adding the digits 9 and 3 overflows our maximum digit value 9, we carry over one digit to the left.
The part of the calculated numbers greater than 10 is our calculated digit at
the current position. For 9 + 3 = 12, the amount greater than 10 is 2,
therefore, 2 is the resulting digit at the right-most position.
Step 2: (Move one step to the left, add digits)
1 + 0 + 0 = 1
The first number in this calculation is the carry, the rest of the digits are 0, since no second digit is present in either 9 or 3.
Since 1 is not greater than 9, we can immediately write down 1 as the second digit.
Another example
64182 + 5499 = 69681
| Carry → | 1 | 1 | ||||
|---|---|---|---|---|---|---|
| 6 | 4 | 1 | 8 | 2 | ||
| 5 | 4 | 9 | 9 | |||
| + | ||||||
| Result → | 6 | 9 | 6 | 8 | 1 |
Carry is 0 initially.
carry + 2 + 9 = 11, carry = 1, digit: 1carry + 8 + 9 = 18, carry = 1, digit: 8carry + 1 + 4 = 6, carry = 0, digit: 6carry + 4 + 5 = 9, carry = 0, digit: 9carry + 6 + 0 = 6, carry = 0, digit: 6
Your calculator will implement the high school addition algorithm for positive numbers. Your function should be provided two integer parameters and return an integer.
Implement the high school addition algorithm, you may choose to use the function sum in code/arith.py as a starting point. In addition to providing a starting point for the function structure, the function also makes use of the functions get_digit and get_number_length. The first function returns the n-th digit from the right (starting at 0) of number. The second function returns the number of digits for number. A third function set_digit may be used to set the n-th digit of number to the digit v.
The next function that will be implemented is multiplication using the
previously implemented function for adding two positive integers. i.e. Let's
suppose that we want to find the outcome of 5 * 5, we will be able to write
this as 5 + 5 + 5 + 5 + 5. In other words, we can write the expression a * b
as adding a b times.
Your calculator will implement multiplication for positive numbers by making use of the addition function. Your function should be provided two integer parameters and return an integer.
You make choose to use the function multiply in code/arith.py as a starting point.
Similar to the algorithm for adding two numbers, we can implement functionality for calculating the subtraction of two numbers.
For the subtraction version of the algorithm, most steps will be equal. We will start at the right, processing digits and moving one digit to the left afterwards. Instead of adding digits, however, we will now subtract digits. And, instead of computing the carry, we will now borrow from the digit to the left in case the result of subtracting the digits is less than 0. In case we borrow from the next digit, we add 10 to the calculation of the subtraction of the two digits.
An example
64182 - 5499 = 58683
| Borrow → | 0 | 1 | 1 | 1 | 0 | |
|---|---|---|---|---|---|---|
| 6 | 4 | 1 | 8 | 2 | ||
| 5 | 4 | 9 | 9 | |||
| - | ||||||
| Result → | 5 | 8 | 6 | 8 | 3 |
Borrow is 0 initially.
2 - 9 - borrow = -7, borrow = 1, digit: 38 - 9 - borrow = -2, borrow = 1, digit: 81 - 4 - borrow = -4, borrow = 1, digit: 64 - 5 - borrow = -2, borrow = 1, digit: 86 - 0 - borrow = 5, borrow = 0, digit: 5
As we can see, when we calculate two digits and need to borrow from the digit to the left:
2 - 9 - borrow = -7We determine borrow to be 1, as the result of subtracting the digits is less than 0. This requires us to add 10 (the amount that we borrowed from the next digit), to be added to the result. Which yields-7+10=3as the outcome of subtracting the digits at the right-most position.
Your calculator will implement the high school subtraction algorithm for positive numbers. Your function should be provided two integer parameters and return an integer.
You make choose to use the function sub in code/arith.py as a starting point.
The last function that will be added to our calculator is the division of two
positive integers. Similar to multiplication, division can also be implemented
using the functionality that we've implemented earlier in this assignment. For
division, we will be able to make us of the subtraction of two positive
integers. i.e. Let's suppose that we want to calculate the outcome of the
expression 25 / 6, we will be able to write this as 25 - 6 - 6 - 6 - 6, such
that the outcome of the latter expression is smaller than 6. In other words, we
keep subtracting the divisor until we cannot subtract anymore. In this case,
we would find the outcome to be 4, since 4 * 6=24. We will only be dealing
with integers.
Implement the multiplication function. Your function should be provided two integer parameters and return an integer.
You make choose to use the function divide in code/arith.py as a starting point.
Thus far, we have limited ourselves to positive numbers. We can, however, extend our implementation to also support negative numbers. Moreover, we can even implement subtraction by means of addition, i.e. a - b = a + -b.
For refinement of the arithmetic functionality implemented this week, implement support for negative numbers. Please refer to the final assignment description criteria for the appropriate format such that your final solution can be correctly evaluated.
As a reference, you can use the following table:
| Variables | a, b | -a,-b | +a,-b | -a, +b |
|---|---|---|---|---|
| Addition | (+a)+(+b)=a+b | (-a)+(-b)=(-1)*(a+b) | (+a)+(-b)=a-b | (-a)+(+b)=b-a |
| Subtraction | (+a)-(+b)=a-b | (-a)-(-b)=(-a +b)=b-a | (+a)-(-b)=a+b | (-a)-(+b)=(-1)*(a+b) |