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+[Contents](../Contents.md) \| [Previous (1.2 A First Program)](02_Hello_world.md) \| [Next (1.4 Strings)](04_Strings.md)
+
+# 1.3 Numbers
+
+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`.
+
+```python
+a = True
+b = False
+```
+
+Numerically, they're evaluated as integers with value `1`, `0`.
+
+```python
+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:
+
+```python
+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:
+
+```python
+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](https://en.wikipedia.org/wiki/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.
+
+```python
+>>> 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.
+
+```python
+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:
+
+```python
+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:
+
+```python
+a = int(x)    # Convert x to integer
+b = float(x)  # Convert x to float
+```
+
+Try it out.
+
+```python
+>>> a = 3.14159
+>>> int(a)
+3
+>>> b = '3.14159' # It also works with strings containing numbers
+>>> float(b)
+3.14159
+>>>
+```
+
+## Exercises
+
+Reminder: These exercises assume you are working in the `practical-python/Work` 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:
+
+```python
+# 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`.
+
+### 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 `929,965.62` 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:
+
+```python
+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:
+
+```bash
+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 880074.1 -1871.53
+Total paid 880074.1
+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,
+
+```python
+>>> int("123")
+123
+>>> float("1.23")
+1.23
+>>>
+```
+
+With that in mind, can you explain this behavior?
+
+```python
+>>> bool("False")
+True
+>>>
+```
+
+[Contents](../Contents.md) \| [Previous (1.2 A First Program)](02_Hello_world.md) \| [Next (1.4 Strings)](04_Strings.md)