Some math expressions look strange until you connect them to something real.
Here are a few that make more sense with the right picture in mind.
0/0 (Zero divided by zero)
Zero divided by zero is not zero, but undefined.
Let's think of a law enforcement agent working with an informant, and receiving regular tips:
- Week 1: The agent gets 4 tips from the informant, and decides to raid all 4 places. Finds something in 2 of them. Success rate of the informant is:
2/4 = 0.5. - Week 2: The agent gets 3 tips, and again decides to raid all, but finds nothing. Success rate of the informant can be said to be
0/3 = 0. - Week 3: The agent gets 5 tips, but this time decides to raid none of them. Success rate:
0/0.
This last case is different. No raids mean no chance to measure success or failure.
We can't say the informant was bad (0 success), and we can't say good either.
That's why 0/0 is undefined, not zero.
0! (Zero factorial)
Zero factorial is not 0, but 1.
The factorial is a fundamental concept in combinatorics, the study of counting.
The number of ways to arrange n distinct objects is given by n!.
For example:
There are 3! = 6 ways to arrange three different books on a shelf.
There are 2! = 2 ways to arrange two different books.
There is 1! = 1 way to arrange one book.
Now, how many ways are there to arrange zero books?
If you have a set with zero objects (an empty set), there is only one way to arrange them: by doing nothing.
The "arrangement" is the empty shelf arrangement itself.
So, the number of ways to arrange zero objects is 1.
Therefore, 0! = 1. This interpretation is the one most commonly used in mathematics to define 0!.
i (Unit imaginary number)
Unit imaginary number, the square root of -1, corresponds to 90 degrees in 2D vector space.
But it isn't so imaginary if you think of it as a rotation.
First consider multiplying a number by -1. This changes its direction by 180 degrees on the number line.
Multiplying a number by i, corresponds to rotating it 90 degrees.
Doing this again results in a 180 degrees rotation.
That's the same as flipping to the negative side: i × i = -1.
This is the geometric reason why i = √-1.
0⁰ (Zero to the power of zero)
At first glance, 0 raised to the power of 0 looks like nonsense.
One may be tempted to say it is 0.
Particularly in mathematical analysis, it is often considered an indeterminate form.
But in combinatorics and algebra it is actually just 1.
This one may not sound as intuitive as the ones above, but let's start with a foundational combinatorial principle:
If we have a set A with k elements and a set B with n elements, the number of possible functions (or mappings) from set A to set B is given by nᵏ.
When we say "number of possible functions" we mean: how many different ways can we assign each element of one set (the domain) to elements of another set (the codomain).
For example:
- Domain =
{Alice, Bob}(2 people). - Codomain =
{Coffee, Tea, Juice}(3 drinks). - Alice can pick any of the 3 drinks.
- Bob can also pick any of the 3 drinks, no matter what Alice chose.
- That gives
3² = 9different "assignment rules".
Each of those 9 is a distinct function (one might map both names to Coffee, another maps Alice to Tea, Bob to Juice, etc.).
But at the end there 9 different functions can exist for this mapping. Let's list all those functions actually:
- Function 1:
Alice → Coffee, Bob → Coffee - Function 2:
Alice → Coffee, Bob → Tea - Function 3:
Alice → Coffee, Bob → Juice - Function 4:
Alice → Tea, Bob → Coffee - Function 5:
Alice → Tea, Bob → Tea - Function 6:
Alice → Tea, Bob → Juice - Function 7:
Alice → Juice, Bob → Coffee - Function 8:
Alice → Juice, Bob → Tea - Function 9:
Alice → Juice, Bob → Juice
Each of these functions would take a name as input and return a drink as output.
And as you can see there 3² = 9 distinct functions. No more, no less.
Now let's apply this to the case of 0⁰.
We need to consider the number of functions from a set of size 0 to a set of size 0.
- Domain: is a set with 0 elements, which is the empty set, denoted by
∅. - Codomain: is also the empty set,
∅. - So, we are trying to find the number of functions
f: ∅ → ∅.
A function is a rule that assigns each element of the domain to exactly one element of the codomain.
Let's follow this rule for our two empty sets:- "...assigns each element of the domain..."
- The domain is the empty set. It has no elements.
- This condition is satisfied because there are no elements to assign. The rule "for each element" is trivially true because there are no elements to check.
- "...to exactly one element of the codomain."
- This part of the rule is also satisfied, again because there are no elements to assign.
- "...assigns each element of the domain..."
So, the "rule" of the function holds true. The assignment is just a "do nothing" assignment. There is exactly one way to do nothing.
The "one way to do nothing" is the empty function.
Since there is only one such function (the empty function itself), the number of ways to map elements from a set of 0 elements to a set of 0 elements is 1. Therefore, 0⁰ = 1.
Negative exponent
Exponents don't stop at whole numbers. Once you see the pattern, negatives and fractions fit naturally.
a⁻¹ just means "divide by a".
Because a¹ × a⁻¹ = a⁽¹⁻¹⁾ = a⁰ = 1.
Example: 2⁻¹ = 1/2, 10⁻¹ = 1/10.
In general: a⁻ⁿ = 1 / (aⁿ).
This comes directly from the rule aᵐ × aⁿ = a⁽ᵐ ⁺ ⁿ⁾.
If you want exponents to add consistently, negative exponents must represent division.
Fractional exponent
a¹ᐟ² means "a number that when multiplied by itself gives a".
Because a¹ᐟ² × a¹ᐟ² = a⁽¹ᐟ² ⁺ ¹ᐟ²⁾ = a¹ = a.
That's why a¹ᐟ² is the square root.
Similarly, a¹ᐟ³ means "a number that when multiplied by itself three times gives a".
Because a¹ᐟ³ × a¹ᐟ³ × a¹ᐟ³ = a⁽¹ᐟ³ ⁺ ¹ᐟ³ ⁺ ¹ᐟ³⁾ = a¹ = a.
That's why a¹ᐟ³ is the cube root.
More generally:
a¹ᐟⁿ is the n-th root of a.
aᵐᐟⁿ = (a¹ᐟⁿ)ᵐ.
Example: 8²ᐟ³ = (8¹ᐟ³)² = 2² = 4.
