De Moivre's formula
De Moivre's formula is a way to find powers of complex numbers. But, before we can understand that, first we need to know what complex numbers are.
Complex numbers are numbers that have two parts: a real part and an imaginary part. The imaginary part is represented by the letter 'i'. So, when we see a complex number like 3 + 2i, that means the real part is 3, and the imaginary part is 2 times i.
Now, De Moivre's formula comes into play when we want to find what a complex number raised to a power is. To do that, we need to use angles.
Think of a complex number as a point on a graph. The real part is the position on the X-axis, and the imaginary part is the position on the Y-axis. So, if we have a complex number 3 + 2i, it would be a point 3 units to the right of the origin and 2 units up from the origin.
Now, draw a line from the origin to that point. This line forms an angle with the X-axis. That angle is what we call the argument of the complex number.
So, when we want to find a complex number raised to a power, we need to figure out the argument of that number. Then, we can use De Moivre's formula to find out what the number raised to that power is.
The formula is: (cosine(angle) + i * sine(angle))^n
Let's break that down. First, we need to find the cosine and sine of the angle. Luckily, we can use math tools to do that.
Once we've calculated the cosine and sine of the angle, we can put those values into the formula. Then, we raise that to the power we want.
The result we get will be a new complex number. It will have a real part and an imaginary part, just like before. And, it will be the original complex number raised to the power we wanted.
So, in simpler terms, De Moivre's formula helps us find what complex numbers raised to a power are. We have to figure out the angle of the complex number, use the formula with cosine and sine, and raise it to the power we want.
Complex numbers are numbers that have two parts: a real part and an imaginary part. The imaginary part is represented by the letter 'i'. So, when we see a complex number like 3 + 2i, that means the real part is 3, and the imaginary part is 2 times i.
Now, De Moivre's formula comes into play when we want to find what a complex number raised to a power is. To do that, we need to use angles.
Think of a complex number as a point on a graph. The real part is the position on the X-axis, and the imaginary part is the position on the Y-axis. So, if we have a complex number 3 + 2i, it would be a point 3 units to the right of the origin and 2 units up from the origin.
Now, draw a line from the origin to that point. This line forms an angle with the X-axis. That angle is what we call the argument of the complex number.
So, when we want to find a complex number raised to a power, we need to figure out the argument of that number. Then, we can use De Moivre's formula to find out what the number raised to that power is.
The formula is: (cosine(angle) + i * sine(angle))^n
Let's break that down. First, we need to find the cosine and sine of the angle. Luckily, we can use math tools to do that.
Once we've calculated the cosine and sine of the angle, we can put those values into the formula. Then, we raise that to the power we want.
The result we get will be a new complex number. It will have a real part and an imaginary part, just like before. And, it will be the original complex number raised to the power we wanted.
So, in simpler terms, De Moivre's formula helps us find what complex numbers raised to a power are. We have to figure out the angle of the complex number, use the formula with cosine and sine, and raise it to the power we want.