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I was reading up on the properties modulo function, when I saw the property:

$$-a equiv (10-a) space pmod10$$

Which means

$$-1 equiv (10-1) equiv 9 space pmod10$$

Now:
$$iota = sqrt-1$$

Substituting $-1 equiv 9 pmod10$:

$$iota equiv sqrt9 pmod10$$
$$iota equiv pm 3 pmod10$$

Where did I go wrong?

There are a few things going on in this case.

• Mod typically deals in non-negative integers, they just have negative equivalents.


• Mod only cares, if it can exist in the integers.


• Every perfect square ending in 9 ( aka -1 mod 10) is a square of a number that ends in 3 or 7 ( aka 3 and -3 mod 10)


• iota is not the square root of -1 ...


• If a statement has an example, it will work in any Mod, not just Mod 10.


• 
  

  Some statements only work in one direction.

  
  
  

  Thanks for learning some basic Mod rules.

  
  

As others already mentioned, the problem is that $i$ is not an integer, so the typical modular arithmetic doesn't really make sense here. However, there's a way to get something similar, by working with the gaussian integers $mathbb Z[i]$, which is the set of complex numbers with integer comoponents together with the usual addition and multiplication. Suppose we take the principal ideal $(10)subseteqmathbb Z[i]$, then we look at the image of the elements of $mathbb Z[i]$ under the natural ring homomorphism $phi:mathbb Z[i]tomathbb Z[i]/(10)$ (here, we are "looking at the gaussian integers mod $10$"). We do indeed have
$$ i^2=-1=9=3^2=7^2pmod10. $$
But the problem with concluding that $i=3$ or $7$ is that in general, $x^2=y^2$ does not imply $x=y$. It does not even imply $x=pm y$. That's only true in the field of real numbers. To take a much easier example, note that
$$ 1^2=3^2=5^2=7^2=1pmod8,$$
but obviously, $1,3,5,7$ are not equal.

The confusion in the question is in thinking that $sqrt-1$ always denotes the same number. In the end, the expression "$sqrt-1$" denotes a number whose square is $-1$. So the meaning of the symbol $sqrt-1$ depends on the multiplication operation - if we switch to a different multiplication operation, we can get a different meaning for some square roots.

• In the real numbers, there is no value for $sqrt-1$ - there is no real number that is the square root of $-1$ under multiplication of real numbers.




• In the complex numbers, there are two possible values for the square root of $-1$ under multiplication of complex numbers.




• If we work with integers modulo $5$, $2 = sqrt-1$, because $2^2 = 4 equiv -1 pmod5$. But $4$ is not a square root of $-1$, modulo $5$, because $4^2 = 16 equiv 1 not equiv -1 pmod5$.




• If we work with integers modulo $17$, now we have $4 = sqrt-1$, because $4^4 = 16 equiv -1 pmod17$.




• If we work modulo 3, there is no square root of $-1$, because $1^2 equiv 2^2 equiv 1 not equiv -1 pmod3$.

There is nothing special about square roots here. For example, modulo $5$ we have $1/2 = 2^-1 equiv 3 pmod5$, because $3$ multiplied by $2$ gives $1$, modulo $5$. Of course the real numbers $1/2$ and $3$ are different, but when we work with a different multiplication operation the meaning of fractions can be different.