The relationship between two things is something that we learn early in life. It's quite an important thing to know about, especially for understanding the world we live in.
However the question of 'are these the same' or 'are these equal' is surprisingly tricky. It might immediately seem rather obvious, but in actual fact there is a surprising amount of complex mathematics to do with equality and how you define it. As it turns out there are quite a few ways of saying that things are the same, leading to a description of all the different ways of comparing things being called an Equivalence relationship.
Equivalence relationships have three key properties:
- Reflexive - That everything is equal to itself.
- Symmetric - That if object A is equal to thing B then thing B must be equal to object A.
- Transitive - That if A equals B and B equals C then A must equal C.
For example consider the following:
Where it starts to get tricky is if there is some variation between the two things being compared. Initially the answer to this might appear to be "no, they aren't equal", but consider the following:
- Yes, it's the same image.
- Yes, apart from one is a bit distorted.
- No, they look different.
This all ties in very deeply with one of the foundations of computer science; the distinction between comparing the concept represented by some data and the data itself. A clearer example of the distinction here would be to use two pictures where the idea of equality is even more fuzzy.
As before the answer to if these two are equal is yes and also no. If the question being asked is "are these the same cat?" then the answer is yes; if the question is instead "are these photos the same?" then the answer is no. Usually it's obvious which of these questions is being asked from the context and generally the question is phrased in the more explicit way.
Computers are no different. Ever since Ada Lovelace realised that numbers could be used to represent ideas, concepts and physical objects there has been the need to distinguish between the two types of equality. It's necessary to explicitly tell the computer if you're asking if the numbers are the same (asking if the photos are the same for the above example) or asking if the thing the numbers represent are the same (asking if it's the same cat).
This distinction between the values and what the values are meant to represent occurs frequently in computer languages. For example in the Java language the operator '==' is used to compare the number representing the object, where as the 'equals()' method is used to compare the concept/item that the object represents.
This distinction between a character/digit of 5 and the number 5 (which you get from adding 2 and 3) might seem a little strange, but consider the phrase "I ate 5 strawberries". It's necessary for there to be some way to represent each character/letter as a number and the method of representation that has become standard has the digit 5 represented by the number 53.
So the next time you're trying to work out if two things are the same, just remember to think what you are wanting to compare.