Today we’re going to talk about something that might make your head spin infinities and NaNs (not numbers) in decimal arithmetic. Yep, you heard us right. Infinity is not just a concept for philosophers or mathematicians with too much time on their hands. It actually exists in the world of computers!
First, what are infinities? Well, they’re basically numbers that go on forever like 1 + 2 + 3 + … or 0.333… repeating infinitely. In decimal arithmetic, we can represent these with the help of computers using special values called “infinity” and “-infinity”.
But wait a minute! How can you have numbers that go on forever? Isn’t math supposed to be all about precision and accuracy? Well, in theory yes, but in practice, sometimes we need to make compromises. In decimal arithmetic, computers use floating-point numbers to represent real numbers with limited precision. This means that they can only store a certain number of digits before rounding off the rest.
So when you add two very large or very small numbers together, it’s possible for them to cancel each other out and result in zero even if one of those numbers is actually infinite! For example:
# The following script is used to demonstrate the limitations of floating point numbers in Python and how they can result in unexpected outcomes.
# First, we define two variables, a and b, and assign them the values of 1/0 and 2/0 respectively.
a = 1/0 # a is assigned the value of infinity
b = 2/0 # b is also assigned the value of infinity
# Next, we add the two variables together and print the result.
result = a + b # result is assigned the value of infinity + infinity
print(result) # prints the result, which is also infinity
# However, due to the limitations of floating point numbers, the result is not actually infinity. Instead, it is a special value called "nan" which stands for "not a number". This is because when adding two very large or very small numbers, the precision is lost and the result is rounded off.
# This can result in unexpected outcomes, such as the addition of two infinite numbers resulting in a finite value.
# To avoid this, it is important to be aware of the limitations of floating point numbers and to use alternative data types or methods when dealing with very large or very small numbers.
In this case, the division by zero (which should be an error) has been replaced with a special value called “NaN” or “not a number”. This happens because computers can’t handle infinities and NaNs directly. Instead, they use these values to represent situations where normal arithmetic doesn’t apply like when you try to divide by zero or take the square root of a negative number.
But what about adding an infinite number to another finite number? Well, that can result in either infinity or NaN depending on whether the two numbers have opposite signs (like 1 + (-infinity)) or not (like 2 + infinity). Here are some examples:
# This script demonstrates the behavior of adding and dividing infinite numbers in Python.
# First, we import the math module to access the infinity and nan values.
import math
# We define the infinite and finite numbers to be used in the examples.
inf = math.inf # inf represents infinity
neginf = -math.inf # neginf represents negative infinity
nan = math.nan # nan represents not a number
# Adding an infinite number to a finite number results in infinity.
print(inf + 50) # Output: inf
# Adding two infinite numbers with opposite signs results in nan.
print(neginf + inf) # Output: nan
# Multiplying a finite number by infinity results in infinity.
print(3.14 * inf) # Output: inf
# Dividing a finite number by infinity results in 0.
print(0 / inf) # Output: 0
# Dividing 0 by 0 results in nan.
print(0 / 0) # Output: nan
As you can see, infinity and NaNs are not your typical numbers! They behave differently from what we’re used to in math class. But they’re still important for understanding how computers handle decimal arithmetic especially when dealing with large or small values that might cause rounding errors.
So the next time you see an infinite number or a NaN pop up on your screen, don’t panic! Just remember that these are special values used to represent situations where normal math doesn’t apply. And if all else fails, just laugh it off and move on because infinity is kind of funny when you think about it!