Why does i2=−1?

Question

Why does i2=−1?

Maricela Alarcon · Accepted Answer

This question really isn’t any different from the question “why does

3^{2} = 9 ?

''. Both “why does

i^{2} = - 1 ?

and “why does

3^{2} = 9 ?^{″}

are answered with “this follows from the definition of x” (replacing x with ‘i’ or '3’ as appropriate).
If the question now goes further and asks: “but why is i defined in that way?”, then the answer is somewhat blunt:
You don’t question definitions.
It would be like questioning “why is the queen piece in chess called a ‘queen’?”. Or, for that matter, “why is 3 defined in that way?” The answer is that we decided to encapsulate the properties of the number three, say, into the symbol ‘3’. In short, the properties of the number three came first, and this set of properties was assigned the symbol ‘3’ in order to be simpler for us to pursue the study of these properties further. Not vice-versa. The number 3 did not come before its properties, but after.
Likewise for i . At some point, someone thought it was a good idea to look into the properties of numbers satisfying equations such as

x^{2} + 1 = 0 or x^{2} + 2 x + 5 = 0

.
Thus, the definition of i was born. But the properties of i were present before its nomenclature, not after; indeed, this definition was made in order for these properties to be further explored.

Jacob Homer · Accepted Answer

Instead, it came down to starting with i2=−1, and then trying to break it. If it couldn’t be broken, then it needed to be well-described. This is what mathematicians did.   Someone thought about it and said, “I’ll just use i here and see what happens”.   That’s called a construction. Someone was interested in the implications of what is left over by forming this construction, and comparing those to the rules of other algebras and fields. Careful application of logic and painstaking discipline went into determining if i2=−1 led to logical inconsistencies, and in the end, it didn’t, but it did lead to some amazing discoveries indeed.

alenahelenash · Accepted Answer

We define the imaginary unit i to be one of the two numbers satisfying i2=−1. Whichever one we choose (because it ultimately doesn&#039;t matter), the other is denoted -i.

Why does i^{2}=-1?

Answered question

Answer & Explanation

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