Mathematical Calculations Tutorial

Introduction

This section discusses mathematical calculations.

Types of Numbers

Python has 4 types of numbers:

• Booleans
• Integers
• Floating point
• Complex (imaginary numbers)

Booleans (bool)

Booleans have two values: True, False.

a = True
b = False

Numerically, they're evaluated as integers with value 1, 0.

c = 4 + True ## 5
d = False
if d == 0:
    print('d is False')

But, don't write code like that. It would be odd.

Integers (int)

Signed values of arbitrary size and base:

a = 37
b = -299392993727716627377128481812241231
c = 0x7fa8      ## Hexadecimal
d = 0o253       ## Octal
e = 0b10001111  ## Binary

Common operations:

x + y      Add
x - y      Subtract
x * y      Multiply
x / y      Divide (produces a float)
x // y     Floor Divide (produces an integer)
x % y      Modulo (remainder)
x ** y     Power
x << n     Bit shift left
x >> n     Bit shift right
x & y      Bit-wise AND
x | y      Bit-wise OR
x ^ y      Bit-wise XOR
~x         Bit-wise NOT
abs(x)     Absolute value

Floating point (float)

Use a decimal or exponential notation to specify a floating point value:

a = 37.45
b = 4e5 ## 4 x 10**5 or 400,000
c = -1.345e-10

Floats are represented as double precision using the native CPU representation IEEE 754. This is the same as the double type in the programming language C.

17 digits of precision
Exponent from -308 to 308

Be aware that floating point numbers are inexact when representing decimals.

>>> a = 2.1 + 4.2
>>> a == 6.3
False
>>> a
6.300000000000001
>>>

This is not a Python issue, but the underlying floating point hardware on the CPU.

Common Operations:

x + y      Add
x - y      Subtract
x * y      Multiply
x / y      Divide
x // y     Floor Divide
x % y      Modulo
x ** y     Power
abs(x)     Absolute Value

These are the same operators as Integers, except for the bit-wise operators. Additional math functions are found in the math module.

import math
a = math.sqrt(x)
b = math.sin(x)
c = math.cos(x)
d = math.tan(x)
e = math.log(x)

Comparisons

The following comparison / relational operators work with numbers:

x < y      Less than
x <= y     Less than or equal
x > y      Greater than
x >= y     Greater than or equal
x == y     Equal to
x != y     Not equal to

You can form more complex boolean expressions using

and, or, not

Here are a few examples:

if b >= a and b <= c:
    print('b is between a and c')

if not (b < a or b > c):
    print('b is still between a and c')

Converting Numbers

The type name can be used to convert values:

a = int(x)    ## Convert x to integer
b = float(x)  ## Convert x to float

Try it out.

>>> a = 3.14159
>>> int(a)
3
>>> b = '3.14159' ## It also works with strings containing numbers
>>> float(b)
3.14159
>>>

Reminder: These exercises assume you are working in the ~/project directory. Look for the file mortgage.py.

Exercise 1.7: Dave's mortgage

Dave has decided to take out a 30-year fixed rate mortgage of $500,000 with Guido's Mortgage, Stock Investment, and Bitcoin trading corporation. The interest rate is 5% and the monthly payment is $2684.11.

Here is a program that calculates the total amount that Dave will have to pay over the life of the mortgage:

## mortgage.py

principal = 500000.0
rate = 0.05
payment = 2684.11
total_paid = 0.0

while principal > 0:
    principal = principal * (1+rate/12) - payment
    total_paid = total_paid + payment

print('Total paid', total_paid)

Enter this program and run it. You should get an answer of 966279.5999999957.

Exercise 1.8: Extra payments

Suppose Dave pays an extra $1000/month for the first 12 months of the mortgage?

Modify the program to incorporate this extra payment and have it print the total amount paid along with the number of months required.

When you run the new program, it should report a total payment of 929965.6199999959 over 342 months.

Exercise 1.9: Making an Extra Payment Calculator

Modify the program so that extra payment information can be more generally handled. Make it so that the user can set these variables:

extra_payment_start_month = 61
extra_payment_end_month = 108
extra_payment = 1000

Make the program look at these variables and calculate the total paid appropriately.

How much will Dave pay if he pays an extra $1000/month for 4 years starting after the first five years have already been paid?

Exercise 1.10: Making a table

Modify the program to print out a table showing the month, total paid so far, and the remaining principal. The output should look something like this:

1 2684.11 499399.22
2 5368.22 498795.94
3 8052.33 498190.15
4 10736.44 497581.83
5 13420.55 496970.98
...
308 874705.88 3478.83
309 877389.99 809.21
310 880074.10 -1871.53
Total paid 880074.10
Months 310

Exercise 1.11: Bonus

While you're at it, fix the program to correct for the overpayment that occurs in the last month.

Exercise 1.12: A Mystery

int() and float() can be used to convert numbers. For example,

>>> int("123")
123
>>> float("1.23")
1.23
>>>

With that in mind, can you explain this behavior?

>>> bool("False")
True
>>>

Summary

Congratulations! You have completed the Numbers lab. You can practice more labs in LabEx to improve your skills.