Today were going to talk about one of the most exciting topics in programming: floating point arithmetic issues and limitations. Yay!
To begin with: what are floating point numbers? They’re a way for computers to represent decimal numbers with a fractional part (like 3.14) using binary digits (bits). This is important because it allows us to do calculations on these numbers much faster than if we were working with decimal numbers all the time.
But heres where things get interesting: floating point arithmetic isn’t perfect. In fact, it can be downright frustrating at times! Let me explain why.
When you perform a calculation using floating point numbers in Python (or any other programming language), there are two main issues to consider: rounding errors and precision limitations.
Rounding Errors: When we do math with decimal numbers, we often have to round the result to get an answer that makes sense for our purposes. For example, if you calculate 1/3 in your head, you might come up with something like .33 or .34 depending on how many digits you want to keep track of.
The same thing happens when we do math with floating point numbers in Python. The problem is that the computer doesn’t know which number you want to round to, so it just rounds to the nearest value it can represent using its limited set of bits. This can lead to some pretty strange results!
For example:
# This script is used to demonstrate the issue with floating point numbers in Python
# First, we declare two variables, x and y, and assign them the values of 0.1 and 0.2 respectively
x = 0.1
y = 0.2
# Next, we add the two variables together and print the result
result = x + y
print(result) # The result should be 0.3, but due to the floating point issue, it will be slightly off
# To fix this issue, we can use the round() function to round the result to the desired number of decimal places
corrected_result = round(result, 1) # The second argument specifies the number of decimal places to round to
print(corrected_result) # The result should now be 0.3, as expected
# Another way to fix this issue is to use the decimal module, which allows for more precise decimal calculations
from decimal import Decimal
# We can declare the variables as Decimal objects instead of regular floats
x = Decimal('0.1')
y = Decimal('0.2')
# Now, when we add the two variables together, the result will be more accurate
result = x + y
print(result) # The result will now be 0.3, as expected
What the ***** is going on here? Well, when you add two numbers that are close together (like 0.1 and 0.2), the computer has to round one of them in order to represent it using its limited set of bits. In this case, Python rounds 0.1 down to 0.0999999999999999847412109375 (which is the closest value that can be represented with a single bit).
When you add these two numbers together, you get something like:
# The following script is used to demonstrate the issue of rounding errors in Python when working with floating point numbers.
# First, we define two variables, x and y, and assign them the values of 0.099999999999999847412109375 and 0.2 respectively.
x = 0.099999999999999847412109375
y = 0.2
# Next, we add these two numbers together and assign the result to a new variable, z.
z = x + y
# Finally, we print the value of z, which should be 0.30000000000000004.
print(z)
# However, due to the limited precision of floating point numbers in computers, the actual value of z is slightly different.
# This is because the computer has to round one of the numbers in order to represent it using its limited set of bits.
# In this case, Python rounds 0.1 down to 0.0999999999999999847412109375, which is the closest value that can be represented with a single bit.
# When we add this rounded value to 0.2, we get the result of 0.30000000000000004.
# To avoid this issue, we can use the round() function to round the numbers to a specific number of decimal places.
# For example, if we round both x and y to 2 decimal places, the result of z will be 0.3.
x = round(x, 2)
y = round(y, 2)
z = x + y
print(z)
This is why you might see strange results when doing math with floating point numbers in Python! The moral of the story: be careful when working with decimal numbers, and always round your answers to a reasonable number of digits.
Precision Limitations: Another issue with floating point arithmetic is that there are limits to how many bits can be used to represent each number. This means that some calculations simply cannot be done accurately using floating point numbers alone!
For example, let’s say you want to calculate the square root of 2 in Python:
# Import the math module
import math
# Use the sqrt function from the math module to calculate the square root of 2
square_root = math.sqrt(2)
# Print the result
print(square_root)
# Output: 1.4142135623730951
# The math module provides access to various mathematical functions, including the sqrt function which calculates the square root of a number.
# The result of the calculation is stored in the variable "square_root".
# The print function is used to display the result to the user.
This is a pretty accurate result, but it’s not exactly what we want! The square root of 2 is actually an irrational number (meaning that its decimal representation goes on forever and never repeats). So why did Python stop after all those digits?
The answer lies in the fact that floating point numbers can only represent a limited set of values. When you calculate the square root of 2, Python has to round the result to the nearest value it can represent using its limited set of bits. This means that we’re never going to get an exact answer when working with irrational numbers!
The moral of the story: be careful when working with irrational numbers or very large/small values in Python, and always check your results for accuracy.