Perform the operation and write the result in standard form.
1 + i over i - 3 over 4 - i
Do I have to multiply top and bottom by i?
Why conjugage?
The conjugage of a given expression is obtained by changing the sign of the imaginary part.
For example, the conjugate of(5x+4i) is (5x-4i), or
the conjugate of (2-3i) is (2+3i).
By multiplying the numerator and denominator by the complex conjugate, the imaginary part of the denominator will be eliminated.
To simplify (i-3)/(4-i), we multiply both the numerator and denominator by the conjugate of (4-i), i.e. (4+i) to give
(i-3)/(4-i)
=(i-3)(4+i)/((4-i)(4+i))
=(i2+i-12)/(42-i2)
=(-1+i-12)/(16-(-1))
=(i-13)/17
For the given problem, you will need to specify more clearly the expression to avoid ambiguity. It could be interpreted as one of the two following cases:
A.
1 + i over i - 3 over 4 - i
= (1+i) / ((i-3)(4-i))
or
B.
1 + i over i - 3 over 4 - i
= (1+i) / ((i-3)/(4-i))
Once the expression is determined, you can proceed to simplify accordingly.
To perform the operation and write the result in standard form, you don't necessarily have to multiply the top and bottom by i. However, doing so will simplify the expression and make it easier to work with.
Let's start by rewriting the expression:
(1 + i) / (i - 3) / (4 - i)
To simplify the expression, you can multiply the top and bottom by the conjugate of the denominator, which is (4 + i). This will eliminate the complex numbers in the denominator.
(1 + i) / (i - 3) * (4 + i) / (4 + i)
Next, we can use the distributive property to multiply the numerators and the denominators:
(1 + i)(4 + i) / (i - 3)(4 + i)
Now let's expand the numerator and denominator:
(4 + i + 4i + i^2) / (4i + i^2 - 12 - 3i)
Simplifying further:
(4 + 5i + i^2) / (i^2 + 4i - 12)
Since i^2 is equal to -1, we can substitute that in:
(4 + 5i - 1) / (-1 + 4i - 12)
Simplifying:
(3 + 5i) / (3 - 4i)
Now, to write the result in standard form, we need to rationalize the denominator. We can do this by multiplying both the numerator and the denominator by the conjugate of the denominator, which is (3 + 4i):
(3 + 5i) / (3 - 4i) * (3 + 4i) / (3 + 4i)
Multiplying the numerators and the denominators:
(9 + 12i + 15i + 20i^2) / (9 + 12i - 12i - 16i^2)
Simplifying further:
(9 + 27i - 20) / (9 + 16)
Simplifying:
(-11 + 27i) / 25
Thus, the result in standard form is -11/25 + (27/25)i.
For
(1+i)/(i-3)
you multiply top and bottom by the conjugate of the denominator, i.e. for i-3, it will be i+3.
Thus
(1+i)/(i-3)
=(i+3)(1+i) / ((i+3)(i-3))
=(i2+4i+3)/(i2+3i-3i-9)
=(-1+4i+3)/(-1-9)
=(4i+2)/(-10)
=-2(2i+1)/(2*5)
=-(2i+1)/5
Try using this technique to apply to your problem.