Why Does i^2 Equal -1? Explained (2024)

I see many people saying that the imaginary number squared is -1, like so:

i = sqrt(-1)
i² = sqrt(-1)*sqrt(-1) = (sqrt(-1))² = -1

But, what about this:

i² = sqrt(-1)*sqrt(-1) = sqrt(-1*-1) = sqrt(1) = 1

Can someone please explain to me why i² = -1 if the above counter example is correct? If not, can someone explain why my counter example is incorrect? It seems like both are correct, but how can that be so?

Thank you

You will understand better when you study Trigonometry.

More simply, we can have a solution for x²+1=0.
x²=-1
x=±√(-1)
and a variable is assigned to this solution, number:
i=±√(-1)

EnlightenedOne said:

I see many people saying that the imaginary number squared is -1, like so:

i = sqrt(-1)
i² = sqrt(-1)*sqrt(-1) = (sqrt(-1))² = -1

Because it does. That's how i is defined, it's the complex number defined such that i^2 = -1.

But, what about this:

i² = sqrt(-1)*sqrt(-1) = sqrt(-1*-1) = sqrt(1) = 1

Can someone please explain to me why i² = -1 if the above counter example is correct? If not, can someone explain why my counter example is incorrect? It seems like both are correct, but how can that be so?

Thank you

In general, the rule of sqrt(a)*sqrt(b) = sqrt(a*b) only works when a and b are both positive reals. That's not the case here.

symbolipoint said:

You will understand better when you study Trigonometry.

More simply, we can have a solution for x²+1=0.
x²=-1
x=±√(-1)
and a variable is assigned to this solution, number:
i=±√(-1)

Actually, I've taken everything up to and including Calc 3, and am currently taking Linear Algebra and ODE. But, we haven't dealt much with complex numbers.

Char. Limit said:

Because it does. That's how i is defined, it's the complex number defined such that i^2 = -1.
In general, the rule of sqrt(a)*sqrt(b) = sqrt(a*b) only works when a and b are both positive reals. That's not the case here.

Oh, ok. I didn't know that sqrt(a)*sqrt(b) = sqrt(a*b) only works for positive real values. So does (sqrt(a))^2 = a work for negative a values? That's what the definition is implying.

Char. Limit said:

Yes it does. That follows from the definition of the square root. I would explain in further detail, but I don't have the background knowledge for that myself. Hopefully another member will.

See Also

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Ok, awesome thank you.

The square root function is discontinuous in nature in the complex plane, so sqrt(x)*sqrt(y) = sqrt(x*y) only applies if x and y are both non-negative and real.

Otherwise you could do this:
-1 = i*i
= sqrt(-1)*sqrt(-1)
= sqrt(-1*-1)
= sqrt(1)
= 1

And now every math guru is feeling a disturbance in the force.

stinsonbr said:

The square root function is discontinuous in nature in the complex plane, so sqrt(x)*sqrt(y) = sqrt(x*y) only applies if x and y are both non-negative and real.

Otherwise you could do this:
-1 = i*i
= sqrt(-1)*sqrt(-1)
= sqrt(-1*-1)
= sqrt(1)
= 1

And now every math guru is feeling a disturbance in the force.

Haha, ok. Thanx!

Defining i as "square root of -1" leads to complications just as shown here. That is why more advanced courses will do the following:
Define the complex numbers to be pairs of real numbers (a, b) with addition defined by (a, b)+ (c, d)= (a+ c, b+ d) and multiplication defined by (a, b)(c, d)= (ac- bd, ad+ bc). We can associate any real number, a, with the pair (a, 0). Then we have (a, 0)+ (b, 0)= (a+ b, 0+ 0)= (a+ b, 0) and (a, 0)(b, 0)= (ab- 0(0), a(0)+ 0(b))= (ab, 0), the usual addition and multiplication of real numbers so we can think of the real numbers as being a subset of the complex numbers.

Of course (0, 1)(0, 1)= (0(0)- 1(1), 0(1)+ 1(0))= (-1, 0), the pair we associate with the real number -1. If we call (0, 1) "i", we are saying that [itex]i^2= (i)(i)= -1[/itex]. Then (a, b)= (a, 0)+ (0, b)= (a, 0) + b(0, 1) so a+ bi.

See Wikipedia's link:
http://en.wikipedia.org/wiki/Imaginary_unit#i_and_.E2.88.92i
http://imageshack.com/a/img909/7874/w7YvFb.jpg
For another analysis see also:
https://www.amazon.com/dp/1441972870/?tag=pfamazon01-20
Joseph Bak,Donald J. Newman
at page 2
amazon has a preview for this page.

Char. Limit said:

Yes it does. That follows from the definition of the square root. I would explain in further detail, but I don't have the background knowledge for that myself. Hopefully another member will.

One method of explanation involves rotation, and can be shown in two-dimensions using a real axis and orthogonal imaginary axis; and this is why I said that Trigonometry can help in understanding. You can try a search on YouTube and expect to find a helpful tutorial lesson on this.

maczenith said:

When I asked my high school trig teacher this question 50 years ago, he told me that functions involving imaginary numbers had to be reduced to their lowest terms. Since sqrt (-1) = i, every time sqrt (-1) appears in the equation, "i" must be substituted. Not a satisfactory answer for a 17 year old. I'm glad to see there's actually more involved.

That was for a FIRST explanation. Later, you learn about complex solutions happening in conjugate pair, which then makes sense.

I think this could be explained without any fuss. Let me explain it to you at the most basic level:

See you have learned that: ##\sqrt(a)\sqrt(b)=\sqrt(ab)##. But I bet you might have never seen your text mentioning something like : ##\sqrt(-a)\sqrt(-b)=\sqrt(-a*-b)=\sqrt(ab)##. It's because its just not true.

See basically great minds defined ##i=\sqrt(-1)##. This actually evolved from the dilemma of "what number when squared gives a negative number". So complex numbers were invented. To be precise 'complex numbers' is a set, and strangely enough real numbers are a subset of this set. Every real number can be defined as a complex number. For instance, '2' can be defined as (2+0i) in complex form (unnecessary formalities!).

So ##\sqrt(-a)## is actually defined as ##\sqrt(-1)\sqrt(a)## where ##\sqrt(-1)=i## hence ##\sqrt(-a)=i\sqrt(a)##.

Therefore ##\sqrt(-a)\sqrt(-b)=\sqrt(-1)\sqrt(a)\sqrt(-1)\sqrt(b)=-1\sqrt(ab)=-\sqrt(ab)##

In the above equation the step which lead to your confusion is used: ##\sqrt(-1)\sqrt(-1)=-1##. (Just accept it, its simply a fact like 2+2=4, you can't prove it, I mean 2+2=4 is because 2+2=4 that's how the universe works. Similarly ##\sqrt(-1)\sqrt(-1)=-1## just because it is like that, nothing else.)
Hope this makes the problem a little clear.

HallsofIvy said:

Defining i as "square root of -1" leads to complications just as shown here. That is why more advanced courses will do the following:
Define the complex numbers to be pairs of real numbers (a, b) with addition defined by (a, b)+ (c, d)= (a+ c, b+ d) and multiplication defined by (a, b)(c, d)= (ac- bd, ad+ bc). We can associate any real number, a, with the pair (a, 0). Then we have (a, 0)+ (b, 0)= (a+ b, 0+ 0)= (a+ b, 0) and (a, 0)(b, 0)= (ab- 0(0), a(0)+ 0(b))= (ab, 0), the usual addition and multiplication of real numbers so we can think of the real numbers as being a subset of the complex numbers.

Of course (0, 1)(0, 1)= (0(0)- 1(1), 0(1)+ 1(0))= (-1, 0), the pair we associate with the real number -1. If we call (0, 1) "i", we are saying that [itex]i^2= (i)(i)= -1[/itex]. Then (a, b)= (a, 0)+ (0, b)= (a, 0) + b(0, 1) so a+ bi.

I don't know how to quote multiple replies on my iPhone mobile browser, so I selected this one.

This post REALLY helps me understand my question! Thank you!

And thanks to all who have replied, your answers helped me also! :D

Hijaz Aslam said:

I think this could be explained without any fuss. Let me explain it to you at the most basic level:
...
Hope this makes the problem a little clear.

Your post is misleading, as you never defined a and b.

GFauxPas said:

Your post is misleading, as you never defined a and b.

GFauxPas - 'a' and 'b' are just variables which define real numbers. I think we often use 'a' and 'b' as common numerical variables in mathematics.

Hijaz Aslam said:

GFauxPas - 'a' and 'b' are just variables which define real numbers. I think we often use 'a' and 'b' as common numerical variables in mathematics.

But you never stated them as positive.

GFauxPas said:

But you never stated them as positive.

GFauxPas - When you say '2', do you take it as a +2 or -2. Of course +2. It's the same here, we often quote a variable 'x','y' or 'z' to represent numerals with the intrinsic direction and 'a','b' and 'c' as positive numerals.

Hijaz Aslam said:

GFauxPas - When you say '2', do you take it as a +2 or -2. Of course +2. It's the same here, we often quote a variable 'x','y' or 'z' to represent numerals with the intrinsic direction and 'a','b' and 'c' as positive numerals.

You are expecting people to assume a variable is a positive number. To communicate formally, better precision is to say what is the kind of number.

Hijaz Aslam said:

GFauxPas - When you say '2', do you take it as a +2 or -2. Of course +2. It's the same here, we often quote a variable 'x','y' or 'z' to represent numerals with the intrinsic direction and 'a','b' and 'c' as positive numerals.

2 is obviously a positive number, but variables that are represented by letters could be positive or negative. We never assume anything about the sign of a variable such as a or x.

symbolipoint said:

You are expecting people to assume a variable is a positive number. To communicate formally, better precision is to say what is the kind of number.

Yes, exactly.