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How Do You Divide Complex Numbers Using Conjugates?

Divide 6 + 3i by 5 + 2i.

Summary

  1. 'i' is the imaginary unit and is equal to the square root of -1
  2. 6 and 5 are the real parts of the complex numbers
  3. 3i and 2i are the imaginary parts of the complex numbers
  4. Multiplying by the complex conjugate, 5-2i, will get rid of the 'i' in the denominator
  5. i2 is equal -1, which is a real number!
  6. Even though 'i' is not a variable, we can treat it like one when we combine like terms

Notes

    1. Fractions are just another representation of division
    2. The dividend, which is the first number in a division problem, goes on the top of the fraction
    3. The divisor, which is the second number, goes on the bottom of the fraction
    4. So for our problem, 6+3i goes on the top and 5+2i goes on the bottom
    1. A 'radical' is another word for a root
    2. Remember, for an expression to be in 'simplest radical form', we can't have any roots in the bottom of a fraction
    1. 'i' is NOT a variable
    2. 'i' is the letter we use to represent the imaginary unit, which is equal to the square root of -1
    3. A 'radical' is another word for a root
    4. If we replaced 'i' with '√(-1)' in our problem, we would have a radical in the denominator!
    5. Remember, in simplest radical form we CANNOT have radicals in the denominator, so that means we can't have any i's either
    1. 'Rationalizing the denominator' means getting rid of any roots in the denominator
    2. To do that, we need to multiply by something that will cancel out the square root
    3. When we did this with regular square roots, we used an expression called a 'conjugate' to cancel the square root
    4. Complex numbers have similar expressions that cancel out imaginary terms
    5. These expressions are called 'complex conjugates'
    1. The denominator of our fraction is 5+2i
    2. Remember, 'i' is equal to the square root of -1
    3. The 'imaginary part' of a complex number is the term with the 'i', so the imaginary part of 5+2i is '2i'
    4. So the complex conjugate of 5+2i is 5-2i
    5. This is just like how we found the conjugate of a radical expression!
    1. When you multiply a complex number by its complex conjugate, it gets rid of the 'i'
    2. So we want to multiply the denominator by its complex conjugate in order to rationalize it
    3. But if we multiply in the denominator, we have to multiply in the numerator as well
    4. If we multiply by something over itself it's the same as multiplying by 1, so we're not changing the value of the fraction
    1. (5-2i)/(5-2i) is just a fancy form of 1
    2. So when we multiply by (5-2i)/(5-2i) we're not changing the value of the fraction
    3. But multiplying 5+2i by 5-2i will cancel out the 'i' term
    4. Then we won't have any radicals in the denominator anymore!
    1. We can multiply complex numbers the same way we multiply regular binomials
    2. Even though 'i' is NOT a variable, we can treat it like one when we multiply
    1. FOIL the complex numbers just like you would two normal binomials
    2. Remember, even though 'i' is not a variable, we can treat it like one when we multiply
    3. Multiply the First terms, 6•5, to get 30
    4. Multiply the Outer terms, 6•(-2i), to get -12i
    5. Multiply the Inner terms, 3i•5, to get 15i
    6. Multiply the Last terms, 3i•(-2i), to get -6i2
    1. FOIL the denominators just like you did with the numerators
    2. Remember, even though 'i' is not a variable, we can treat it like one when we multiply
    3. Multiply the First terms, 5•5, to get 25
    4. Multiply the Outer terms, 5•(-2i), to get -10i
    5. Multiply the Inner terms, 2i•5, to get 10i
    6. Multiply the Last terms, 2i•(-2i), to get -4i2
    1. Multiplying by the complex conjugate should have gotten rid of the 'i' in the denominator
    2. Let's simplify and see if it worked!
    1. Remember, we knew from before that 'i' is equal to the square root of -1
    2. So i2 is equal to the square root of -1 squared
    3. When we square a square root, the square roots cancel out
    4. So that means that i2 is just -1, which is a real number!
    1. Even though 'i' is not a variable, terms with 'i's in them are considered like terms
    2. So again, we can treat 'i' like a variable and combine the 'i' terms
    3. In the numerator, -12i+15i gives us 3i
    4. In the denominator, -10i+10i gives us 0, which cancels out the 'i' terms!
    5. So we DID get rid of the 'i's in the denominator!
    1. Since the 'i' terms canceled out, we're just left with 25+4 in the denominator
    2. 25 and 4 are both real numbers, and they are not radicals, so we've rationalized the denominator!
    3. Now all we need to do is add to finish simplifying!
    4. Adding 30 and 6 gives us 36+3i in the numerator
    5. 25+4 gives us 29 in the denominator
    6. Since 29 doesn't go evenly into 36 or 3, we can't simplify further and our answer is in simplest form!