in this video we're going to talk about imaginary numbers imaginary numbers are basically complex numbers with the imaginary unit i i is equal to the square root of negative 1. i squared is negative one and i to the third is equal to negative i i to the fourth is equal to one now let's talk about why that's the case starting with i to the third power you can think of i to the third power as i squared times i and since i to the second power is negative one i to the third reduces to negative i in the case of i to the fourth think of it as i squared times i squared so it's negative one times negative one which is positive one so with that in mind how would you simplify these imaginary numbers let's say if you have i to the seventh power i to the 26th power i to the 33rd power and i to the 43rd power go ahead and try these what i would recommend doing is breaking up each exponent using the highest multiple of 4. so i to the seventh you can break it up as i to the fourth times i to the third i to the fourth is one i to the third is negative i so i to the seven reduces to negative i now let's move on to the next one i raised to the 26th power so what is the highest multiple of 4 just under 26 well 24 is a multiple of 4. so what i would do is i would write it as i to the 24 times i squared now i to the 24 is basically four times six so it's i to the fourth raised to the sixth power i to the fourth is one if you wait if you raise one to the sixth power it's still going to be one and we can replace i squared of negative one so that's ice to the 26 can be reduced to negative one now let's move on to the next one i to the 33rd power the highest multiple of 4 just under 33 is 32 so we can break up 33 as 32 plus 1 and 32 is 4 times 8. so this is going to be 1 raised to the 8th power times i which is simply i now for the last one i to the 43rd power we can break it up to i to the 40 times i cubed i to the 40 i'm going to write as i to the fourth raised to the 10th power and i to the third is negative i so we can replace i to the fourth with one and so the final answer is just going to be negative i so that's how you could simplify imaginary numbers with a very large exponents now let's talk about adding and subtracting imaginary numbers so let's say you have this problem 5 times 2 plus 3i let's say minus 4 and then times 7 minus 2i how would you simplify this expression feel free to try this problem if you want to in order to simplify this expression the first thing we need to do is distribute let's distribute 5 to 2 plus 3i 5 times 2 is 10 5 times 3i is 15i negative 4 times 7 is negative 28 negative 4 times negative 2i is positive 8i now at this point we need to combine like terms so we can combine 10 and negative 28 that's negative 18 and then 15i and 8i that's going to be positive 23i so at this point our answer is in standard form a plus bi a is the real part of the complex number so is b but combined this makes up the imaginary part b times i even though b is a real number but combine b times i is an imaginary number so that's the solution for this particular problem now let's work on this example so here we're going to multiply two imaginary numbers together so we need to foil 5 times 8 is 40. 5 times 3i that's going to be 15i and then negative 2i times 8 that's negative 16i and then negative 2i times 3i that's negative 6 i squared so all we can do right now is combine like terms and then simplify now i squared is negative 1. so 6 negative 6i squared is going to be positive 6 and now we can combine these two and so we're going to get 46 minus i so when simplifying imaginary numbers or if you're multiplying or dividing you always want to put your final answer in complement in standard form a plus bi form so in this example a is 46 b is the number in front of i which is negative 1. so you can write it as 46 minus 1i if you want now let's try this one how can we simplify this expression 3 plus 2i divided by 4 minus 3i so what do we need to do here when dividing complex numbers one of the best things to do is to multiply the denominator by the conjugate of the complex number so the conjugate of 4 minus 3i is simply 4 plus 3i you just need to change the sign between the real and imaginary number now whatever you do to the bottom you must also do to the top so we've got to multiply both the numerator and the denominator by the conjugate of the denominator now we need to foil 3 times 4 is 12. 3 times 3i that's going to be 9i and then 2i times 4 that's 8i 2i times 3i that's going to be plus 6i squared on the bottom we have 4 times 4 which is 16. well we can get rid of the parenthesis next is going to be 4 times 3i that's going to be plus 12i and that's going to cancel with negative 3i times 4 which is negative 12i and then negative 3i times positive 3i that's negative 9 i squared so now let's simplify 9i plus 8i is 17i and 6i squared is negative 6. these two will cancel negative nine i squared i squared is negative one so negative nine i squared is gonna be plus nine on top we can combine twelve minus six which is six and on the bottom sixteen plus nine is twenty-five so right now we can write the answer as 6 over 25 plus 7 i mean 17 over 25 times i so this is the solution in standard form so we can see that a is 6 over 25 b is 17 over 25. now what would you do if you were to see a problem like this how would you simplify this problem notice that we don't have an a value in the denominator fraction we only have the b value attached to i in a situation like this the best thing to do is to multiply the top and the bottom by i seven times i is seven i two i times i is gonna be two i squared on the bottom we're gonna have five i squared now two i squared that can be rewritten as negative two five i squared is negative 5. so now we could put it in standard form so we have negative 2 divided by negative 5 which is positive 2 over 5 and then positive 7i over negative 5 that's negative 7 over 5i so now we have our answer in a plus bi form a is 2 over 5 b is negative 7 over 5. so you always want to simplify your answer to standard form now sometimes you may need to solve equations associated with complex numbers so let's say we have the equation 4x plus 3i is equal to 12 minus 15y times i so what is the value of x and y in this equation in order to solve this equation we need to identify the real portion of the complex number and the imaginary part on the left side 4x is the real part because it doesn't contain i 12 is the real part as well so we could set 4x equal to 12. on the left side of the equation 3i is the imaginary part and so is negative 15 y i so we could set those two equal to each other on the left we get that x is equal to 12 divided by four or three and on the right to isolate y we need to divide both sides by negative 15 i so we can cancel i and 3 over 15 if you divide both numbers by 3 you can reduce it to negative 1 over 5. and so that is the value of y in this equation so that's how you can solve it now let's say if we want to solve an algebraic equation like this one let's say that x squared plus 36 is equal to zero what is the value of x if you subtract both sides by 36 you'll get that x squared is equal to negative 36 and if you take the square root of both sides you realize that the solution is not a real answer it involves imaginary numbers so what is the square root of negative 36 well we can write the square root of negative 36 as the square root of positive 36 times the square root of negative one the square root of positive 36 is six the square root of negative one is i so we get this answer 6i by the way the square root of 36 you could think of it as plus or minus 6 because we do get two solutions here 6i and negative 6i and we could test it out to make sure it works for instance if we plug in 6i into the equation let's see if it gives us 0 6 squared is 36 i squared is just i squared now we know that i squared is negative 1. so we have 36 times negative 1 which is negative 36. and so we get 0 is equal to 0 this works now if we were to try negative 6i we would also get the same solution because negative 6 times negative 6 is still positive 36 and we're still going to get i squared as well so the end result is still the same thus the solution in this problem is x is plus or minus 6i now let's work on one more example let's plot the complex number 4 plus 3i and at the same time let's calculate the absolute value of 4 plus 3i so let's start with the absolute value so the absolute value of a complex number in standard form is equal to the square root of a squared plus b squared so in this case it's going to be the square root of 4 squared plus 3 squared 4 squared is 16 3 squared is sixteen plus nine is twenty-five so the absolute value is going to be five now let's talk about how we can plot imaginary numbers the x-axis is going to be the real axis the y-axis will be the imaginary axis so the real number is four so we're going to travel four units to the right now the imaginary number combined is 3i so we're going to travel up 3 along the x axis so the point is located here we traveled four units to the right and then up three units the hypotenuse of this triangle represents the absolute value of the complex number and so this is five that's what we have here is the 3 4 5 right triangle and so that's basically it for complex numbers hopefully this video gave you a good introduction in terms of how to add and subtract complex numbers how to multiply how to divide them how to solve equations associated with them how to plot them and also how to find the absolute value of a complex number in standard form