Module 6: Real numbers

Let's start with some math facts:

• Whole numbers like 3, 42 and 1024 are integers.
  (As an aside: integers include 0 and the negative ones like -2 or -219).

• The collection of all integers is infinite in size.

• But integers are limited because some operations on integers do not yield integers:
  • 30 ÷ 5 gives 6, which is an integer.
  • But 31 ÷ 5 is not an integer, yet it's a quantity.

• Real numbers include all the integers but also numbers like 3.141, and -615.2368.

• The collection of all real numbers is also infinite. Interestingly, it's a bigger kind of infinite (but that's a rather subtle math argument outside the scope of this course).

• The term real is just that: a term that's came about historically to describe all these numbers.

• You might wonder: is there any other kind of number?

• Turns out: yes, there a fascinating (and extraordinarily useful) kind of number called an imaginary number, or more generally, a complex number.
  (We won't be working with these in his course.)

• What does one do with real numbers?
  • The same operations: +, -, *, /
  • What's nice is that applying these to real numbers will always result in real number results.

• For example:

  x = 3.14
  y = 2.718
  z = x + y
  print(z)
  
  w = z * (x + y) / (x - y)
  print(w)
      

6.1 Exercise: Type up the above in my_real_example1.py. What is the output?

Note:

• You might have seen 5.8580000000000005 as the value of z printed out.

• However, you might also have seen something slightly different because of the approximate nature of such calculations, a limitation of computer hardware.

• These tiny errors are tiny indeed, but vary slightly from one computer to another, generally occuring around the 16th decimal place: 0.00000000000000001

• Do we need to worry about this? Only if we are engaged in complex scientific calculations.

• Occasionally, however, it can matter. For example, if two people calculate mortgage interest (with real consequences) slightly differently, it could lead to a legal conflict.

Quick review of some relevant math:

• One kind of operation that's useful is power.

• We write \(2^6\) to mean \(2 \times 2 \times 2 \times 2 \times 2 \times 2\) (six times)

• It's easy to see that you could make this work for real numbers that get multiplied: \(2.56^6 = 2.56 \times 2.56 \times 2.56 \times 2.56 \times 2.56 \times 2.56\)

• But could you do \(2^{6.4}\)? Turns out, yes, you can do this even if it's not easy to see or intuit.
  (We would expect \(2^{6.4}\) to be larger than \(2^6\) and smaller than \(2^7\), which it is.)

• The next step then is to allow numbers like \(2.56^{6.4}\).

• In fact, you can take any real number as the mantissa (the 2.56 in \(2.56^{6.4}\)) and any real number as the exponent (the 6.4 in \(2.56^{6.4}\)).

• Let's put this in code and introduce a new operator to raise a number to a power, as in \(2.56^{6.4}\)).

  x = 2 ** 6
  print(x)
  
  y = 2.56 ** 6.4
  print(y)
      

6.2 Exercise: Type up the above in my_real_example2.py. What is the output? Consider \(2.56^x = y\). Can you guess what approximate value of x would make y become 400? Play around with the number 6.4 in the program above and see if you can guess approximately what value would make y become 400.

Let's explore further:

• The technical term for "what is x that would make \(2.56^x = 400\)?" is logarithm.

• We would say: \(x = \log_{2.56}(400)\).
  (Read this as: x is equal to log of 400 to the base 2.56).

• We can calculate this directly:

  import math
  
  x = math.log(400, 2.56)
  print(x)
      

• Here we've introduced some new concepts:

  

• Just like you can ask Python to calculate logarithms using math.log, you can do other kinds of "calculator" functions conveniently.

• Example: math.sqrt for square roots.

6.3 Exercise: In my_real_example3.py, fill in code below

import math

# Write a line of code here

print(x)
      

6.4 Video: