Video Transcript
What is one plus 𝑖 to the power of
10?
Now what we don’t want to do here
is to expand these brackets using a long handed or binomial method. In fact, we’re going to recall De
Moivre’s theorem. Now, De Moivre’s theorem works for
numbers both in exponential and trigonometric form. Let’s look at the exponential
form.
For 𝑧 is equal to 𝑟𝑒 to the
𝑖𝜃, 𝑧 to the power of 𝑛 is equal to 𝑟 to the power of 𝑛 𝑒 to the 𝑖𝑛𝜃. So if we can find a way to convert
our complex number, one plus 𝑖, into exponential form, we can apply De Moivre’s
theorem to work out what one plus 𝑖 to the power of 10 is.
And we can convert a complex number
of the form 𝑎 plus 𝑏𝑖 into exponential form using the formula 𝑟𝑒 to the 𝑖𝜃,
where 𝑟, the modulus, is the square root of 𝑎 squared plus 𝑏 squared. And 𝜃, the argument, is arctan or
inverse tan of 𝑏 divided by 𝑎. For our complex number, one plus
𝑖, 𝑎 is the constant. It’s one. And 𝑏 is the coefficient of
𝑖. It’s also one.
And we can find the modulus then of
our complex number by finding the square root of 𝑎 squared plus 𝑏 squared, which
is the square root of one squared plus one squared. So the modulus is root two. And the argument is the inverse tan
or arctan of one divided by one, which is 𝜋 by four. So we can say that our complex
number, one plus 𝑖, is the same as root two 𝑒 to the 𝜋 by four 𝑖.
And now, we apply De Moivre’s
theorem directly to this complex number. The modulus will be root two to the
power of 10. And the argument will be found by
multiplying 𝜋 by four by 10. Now, if we think about root two as
two to the power of half, we can say that root two to the power of 10 is two to the
power of a half to the power of 10. And we know we need to multiply
these powers. So this is two to the power of
five, which is equal to 32. And 10 multiplied by 𝜋 by four is
10𝜋 by four or five 𝜋 by two.
So we now know the magnitude or the
modulus and the argument of our complex number, one plus 𝑖 to the power of 10. So how do we convert this back into
rectangular form? Well, this comes from the
trigonometric form of a complex number. And we use these two formulae. 𝑎 is equal to 𝑟 cos 𝜃. And 𝑏 is equal to 𝑟 sin 𝜃.
In this case, 𝑎 is equal to 32 cos
of five 𝜋 by two. And 𝑏 is equal to 32 sin of five
𝜋 by two. Cos of five 𝜋 by two is zero. So 𝑎 is equal to zero. And sin of five 𝜋 by two is
one. So 𝑏 is equal to 32.
And since we saw earlier that 𝑏 is
the coefficient of 𝑖, we can say that one plus 𝑖 to the power of 10 is the same as
32𝑖.
