practical-python/Notes/01_Introduction/03_Numbers.md

1.3 Numbers

This section covers some basics of performing mathematical calculations in Python.

Reading

Types of Numbers

Python has 4 types of numbers:

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

Booleans

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')

Don't do that, it would be odd.

Integers

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
x // y     Floor Divide
x % y      Modulo
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

Comparisons

The following comparison / relational operators work with numbers:

<, >, <= >=, ==, !=

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')

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.

17 digits or precision Exponent from -308 to 308

This is the same as the double type in the programming language C.

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

Theses 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)

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)
3.15159
>>>

Exercise 1.3

(a) 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 966,279.6.

(b) 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 929,965.62 over 342 months.

(c) 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 = 60
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 in year 5 of the mortgage?

(d) 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 875705.88 674.44
309 878389.99 -2006.86
Total paid 878389.99
Months 309

(e) Bonus

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

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