Algebra (all content)
- Intro to the imaginary numbers
- Intro to the imaginary numbers
- Powers of the imaginary unit
- Powers of the imaginary unit
- Powers of the imaginary unit
- Simplifying roots of negative numbers
- Simplify roots of negative numbers
- i as the principal root of -1
i as the principal root of -1
The formal definition of i is i^2=-1 and not √-1=i, and there's a good reason for this (although it's a bit technical). See Sal explain why. Created by Sal Khan.
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- 0:10what does principal square root mean?(284 votes)
- When taking square roots of a positive number you always get two answers. For example, the square root of 4 can be either 2 or -2. This is because the square of either one is 4: (2)(2) = 4 = (-2)(-2). So if you want to specify that you mean the positive answer then you would say the "principal square root". So...the principal square root of 4 is 2. The "other" one is -2.(532 votes)
- I am befuddled. So the property functions only when BOTH a and b are positive, and when one of them is negative, the property stays the same, but changes also? So confused...(58 votes)
- The property seems to function when BOTH and and b are NOT negative. If you evaluate the three cases. Case 1: both a and b are positive, Case 2 either a OR b is negative (not both), meaning one is negative, one positive. Case 3 BOTH a AND b are negative.
1) Both a and b are positive. Ex. SQT[4 * 1] = SQT * SQT
SQT[4*1] = SQT=2 (Principal square root only)
2=2 => Success
2) a is positive and b is negative. Ex. SQT[4 * -1] = SQT * SQT[-1]
SQT[4*-1] = SQT[-4]= 2i
2i=2i => Success
3) Both a and b are negative. Ex. SQT[-4 * -1] = SQT[-4] * SQT[-1]
SQT[-4*-1] = SQT=2 (Principal square root only)
SQT[-4]*SQT[-1]=2i*1i=2i^2 (that's 2 i squared)
Since i^2=-1, 2i^2 = -2
2 NOT= -2 => Failure
The property works in Case 1 and Case 2 when at least one of the two are positive. BOTH Negative = failure.
NOTE: SQT is the square root function
* is multiplication
^ is the exponent function(185 votes)
- Wait. I'm lost. How can -1 be equal to 1??(4 votes)
- It's not. What Sal is saying is that people try to prove that i = square root(-1) is wrong because they end up with an answer that 1 = -1 which obviously isn't true. But they're wrong because the square root multiplication rule doesn't apply when both numbers are negative.
Yes the absolute value of -1 = absolute value of 1
but -1 is not equal to one.(118 votes)
- So why exactly can't A and B both be negative? It sounds like they can't because of the "Because I said so" line of reasoning. If √a * √b = √a*b doesn't work when both numbers are negative then maybe something is wrong with the proof.(26 votes)
- Observe what happens.
1 = √1 = √(-1 * -1) = √-1 * √-1 = i * i = -1
Whoops, suddenly, 1 is equal to -1. We broke negative numbers :(
Assuming √(zw) = √z√w leads to contradictions such as above, that's why we ban it.(64 votes)
- What is the square root of i?(21 votes)
- All the imaginary numbers can be written as a+ib where a is the real part and b the imaginary.
Let us assume that i^1/2 = a+ib for some a and b.
i=a^2 + ib^2 + 2aib
ib^2 = -b^2
As the left hand side has only imaginary part, a^2 - b^2 =0
a=b (taking principal roots)
so, i = 2aib
2ab = 1
2a^2 = 1 (a=b)
i^1/2 = (1√2+i 1√2)
Taking 1/√2 common,
i^1/2 = 1/√2(1+i)(1 vote)
- At3:42when it says the radical sign means the principle square root; what's the notation for either square root, or for the negative square root, in that case?
Also when talking about the third root, you wouldn't have to distinguish between principle and negative, right? Do you have to distinguish for all even roots, but not for odd roots?(11 votes)
- The notation for both square roots is ±√x. If you want only the negative square root, then you would say -√x.
As for odd roots, you still have to distinguish between principal and non principal roots. The only difference is that a non principal odd root is not simply the negative of the principal root.
For instance, the cube root of -8 is -2, but it can also be 1±√3 i.(12 votes)
- How can we prove that both A and B cant be negative?(for any values)(7 votes)
- Great question.
*Note that ⁺√ implies principal square root.*
We must prove that:
⁺√(ab) ≠ ⁺√a • ⁺√b
a, b < 0
If a and b are negative, then the square root of them must be imaginary:
⁺√a = xi
⁺√b = yi
x and y must be positive (and of course real), because we are dealing with the principal square roots.
⁺√a • ⁺√b = xi(yi) = -xy
-xy must be a negative real number because x and y are both positive real numbers.
On the other hand,
⁺√(ab) = √[(xi)²(yi)²] = (xyi²)² = (xy)²
Since ⁺√(ab) = (xy)² and ⁺√a • ⁺√b = -xy, our problem becomes to prove that:
(xy)² ≠ -xy
x, y > 0
Well this is easy! The left hand side is obviously positive and the right hand side is obviously negative, so they cannot be equal! Therefore, ⁺√(ab) ≠ ⁺√a • ⁺√b if a, b < 0. Q.E.D. Comment if you have any questions.(17 votes)
- How to distinguish between a radical sign that indicates a principal square root and a radical sign that says the answer must be plus or minus the square root(7 votes)
- Great question.
there is no radical sign that indicates both.
The radical sign indicates the "principal" square root.
That is why when we solve equations like x^2=4 we list two solutions, the principal square root 2 and the negative square root -2. all positive real numbers have two distinct square roots that are opposites of each other.
When solving a problem, if you are looking for square roots, it is up to you to know when you need to consider the negative root. Sometimes it must be included, sometimes it makes no sense and can be discarded.
On the other hand, if you are reading a problem that has the square root symbol in it, it ALWAYS means the principal (positive) root. If the negative is intended, a minus sign will be in front of it. If both are intended they will either be listed separately or the +/- sign will be placed in front of the radical sign. The quadratic formula is usually written with a +/- since you need both the positive and negative roots to find both solutions of a quadratic equation.(3 votes)
- In calculator, if -1, then press square root, why square root of -1 is error on calculator?(13 votes)
- Because real numbers cannot be squared and equal a negative number. (I.e. -3 x -3 = 9). However, imaginary numbers (which are created outside of the normal and "real" numbers) make the square root of -1 possible, but that does not make it "real" or true. Try looking for an "i" sign on your calculator. It can be substituted for the square root of 1.(1 vote)
- why is the principate square root used for negative numbers? can't it be used while multiplying or dividing fractions, however i'm am always asked to simplify the questions but how can i simplify 1/3 times the square root of -63 - the square root of -28?(6 votes)
- Hi Chris,
Remember that you can factor values under the square root symbol. For example, if you are looking for the square root of -63 you can break it up as follows:
√-63 = √9 * √7 * √-1
√9 = 3 and √-1 = i
√-63 = 3(√7)i
If the question is to simplify 1/3 times √-63, then
1/3(√-63) = (3(√7)i)/3
Hope that helps :-)(8 votes)
In your mathematical careers you might encounter people who say it is wrong to say that i is equal to the principal square root of negative 1. And if you ask them why is this wrong, they'll show up with this kind of line of logic that actually seems pretty reasonable. They will tell you that, OK, well let's just start with negative 1. We know from definition that negative 1 is equal to i times i. Everything seems pretty straightforward right now. And then they'll say, well look, if you take this, if you assume this part right here, then we can replace each of these i's with the square root of negative 1. And they'd be right. So then this would be the same thing as the square root of negative 1 times the square root of negative 1. And then they would tell you that, hey, look just from straight up properties of the principal square root function, they'll tell you that the square root of a times b is the same thing as the principal square root of a times the principal square root of b. And so if you have the principal square root of a times the principal square root of b, that's the same thing as the square root of a times b. So based on this property of the radical of the principal root, they'll say that this over here is the same thing as the square root of negative 1 times negative 1. If I have the principal root of the product of two things, that's the same thing as the product of each of their principal roots. I'm doing this in the other order here. Here I have the principal root of the products, over here I have this on the right. And then from that we all know that negative 1 times negative 1 is 1. So this should be equal to the principal square root of 1. And then the principal square root of 1-- Remember, this radical means principal square root, positive square root, that is just going to be positive 1. And they'll say, this is wrong. Clearly, negative 1 and positive 1 are not the same thing. And they'll argue therefore, you can't make this substitution that we did in this step. And what you should then point out is that this was not the incorrect step. That it is true negative 1 is not equal to 1, but the faulty line of reasoning here was in using this property when both a and b are negative. If both a and b are negative, this will never be true. So a and b both cannot be negative. In fact normally when this property is given-- sometimes it's given a little bit in the footnotes or you might not even notice it because it's not relevant when you're learning it the first time-- but they'll usually give a little bit of a constraint there. They'll usually say for a and b greater than or equal to 0. So that's where they list this property. This is true for a and b greater than or equal 0. And in particular, it's false if both a and b are both, if they are both negative. Now, I've just spent the last three minutes saying that people who tell you that this is wrong are wrong. But with that said, I will say that you have to be a little bit careful about it. When we take traditional principal square roots. So when you take the principle square root of 4. We know that this is positive 2, that 4 actually has two square roots. There's negative 2 also is a square root of 4. If you have negative 2 times negative 2 it's also equal to 4. This radical symbol here means principal square root. Or when we're just dealing with real numbers, non imaginary, non complex numbers, you could really view it as the positive square root. This has two square roots, positive and negative 2. If you have this radical symbol right here principal square roots, it means the positive square root of 2. So when you start thinking about taking square roots of negative numbers-- or even in the future we'll do imaginary numbers and complex numbers and all the rest-- you have to expand the definition of what this radical means. So when you are taking the square root of really of any negative number, you're really saying that this is no longer the traditional principal square root function. You're now talking that this is the principal complex square root function, or this is now defined for complex inputs or the domain, it can also generate imaginary or complex outputs, or I guess you could call that the range. And if you assume that, then really straight from this you get that negative, the square root of negative x is going to be equal to i times the square root of x. And this is only-- and I'm going to make this clear because I just told you that this will be false if both a and b are negative. So this is only true-- So we could apply this when x is greater than or equal to 0. So if x is greater than or equal to 0, then negative x is clearly a negative number, or I guess it could also be 0. It's a negative number. And then we can apply this right over here. If x was less than 0, then we would be doing all of this nonsense up here. And we start to get nonsensical answers. And if you look at it this way and you say hey, look, i can be the square root of negative one, if it's the principal branch of the complex square root function. Then you could rewrite this right over here as the square root of negative 1 times the square root of x. And so really the real fault in this logic when people say, hey, negative 1 can't be equal to 1, the real fault is using this property when both a and b, where both of these are negative numbers. That will come up with something that is unambiguously false. If you expand your definition of the complex or expand your definition of the principal root to include negative numbers in the domain and to include imaginary numbers, then you can do this. You can say the square root of negative x is the square root of negative 1 times-- Or you should say the principal square root of negative x-- I should be particular my words-- is the same thing as the principal square root of negative 1 times the principal square root of x when x is greater than or equal to 0. And I don't want to confuse you, if x is greater than or equal to 0, this negative x, that is clearly a negative, or I guess you should say a non positive number.