When we work with complex fractions, one important technique we use is called rationalizing the denominator. This means we’re trying to get rid of any complex numbers in the bottom part of the fraction. We can do this by using something called conjugates. Let’s explore this idea and see how it works!
A conjugate is linked to complex numbers.
If we have a complex number like ( a + bi ) (where ( a ) and ( b ) are real numbers), its conjugate is ( a - bi ).
When you multiply a complex number by its conjugate, you get a real number. Here’s a quick example:
[ (a + bi)(a - bi) = a^2 + b^2 ]
Notice that the imaginary parts cancel out, and we are left with only real numbers.
Sometimes, a complex fraction looks like this:
[ \frac{3 + 4i}{2 - 5i} ]
To make it simpler, especially if we don’t want a complex number in the bottom part, we can use the conjugate of the denominator to help us.
Let’s break it down step-by-step using our example:
Find the Conjugate:
For the denominator ( 2 - 5i ), the conjugate is ( 2 + 5i ).
Multiply by the Conjugate:
We need to multiply both the top (numerator) and the bottom (denominator) by this conjugate:
[ \frac{3 + 4i}{2 - 5i} \cdot \frac{2 + 5i}{2 + 5i} ]
Calculate the New Denominator:
Now, let’s figure out the new bottom part:
[ (2 - 5i)(2 + 5i) = 2^2 + 5^2 = 4 + 25 = 29 ]
Calculate the New Numerator:
Next, we need to calculate the top part:
[ (3 + 4i)(2 + 5i) = 6 + 15i + 8i + 20i^2 = 6 + 23i - 20 = -14 + 23i ]
(Don’t forget that ( i^2 = -1 )!)
Combine Everything:
Now, putting it all together, we have:
[ \frac{-14 + 23i}{29} ]
So, the complex fraction
[ \frac{3 + 4i}{2 - 5i} ]
simplifies to
[ \frac{-14}{29} + \frac{23}{29}i ]
Using the conjugate helps us to get rid of the complex part in the denominator. This makes our calculations easier and helps us with any further steps!
When we work with complex fractions, one important technique we use is called rationalizing the denominator. This means we’re trying to get rid of any complex numbers in the bottom part of the fraction. We can do this by using something called conjugates. Let’s explore this idea and see how it works!
A conjugate is linked to complex numbers.
If we have a complex number like ( a + bi ) (where ( a ) and ( b ) are real numbers), its conjugate is ( a - bi ).
When you multiply a complex number by its conjugate, you get a real number. Here’s a quick example:
[ (a + bi)(a - bi) = a^2 + b^2 ]
Notice that the imaginary parts cancel out, and we are left with only real numbers.
Sometimes, a complex fraction looks like this:
[ \frac{3 + 4i}{2 - 5i} ]
To make it simpler, especially if we don’t want a complex number in the bottom part, we can use the conjugate of the denominator to help us.
Let’s break it down step-by-step using our example:
Find the Conjugate:
For the denominator ( 2 - 5i ), the conjugate is ( 2 + 5i ).
Multiply by the Conjugate:
We need to multiply both the top (numerator) and the bottom (denominator) by this conjugate:
[ \frac{3 + 4i}{2 - 5i} \cdot \frac{2 + 5i}{2 + 5i} ]
Calculate the New Denominator:
Now, let’s figure out the new bottom part:
[ (2 - 5i)(2 + 5i) = 2^2 + 5^2 = 4 + 25 = 29 ]
Calculate the New Numerator:
Next, we need to calculate the top part:
[ (3 + 4i)(2 + 5i) = 6 + 15i + 8i + 20i^2 = 6 + 23i - 20 = -14 + 23i ]
(Don’t forget that ( i^2 = -1 )!)
Combine Everything:
Now, putting it all together, we have:
[ \frac{-14 + 23i}{29} ]
So, the complex fraction
[ \frac{3 + 4i}{2 - 5i} ]
simplifies to
[ \frac{-14}{29} + \frac{23}{29}i ]
Using the conjugate helps us to get rid of the complex part in the denominator. This makes our calculations easier and helps us with any further steps!