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Sections: Introduction, Operations through complexes, The Quadratic Formula

Complex numbers space "binomials" that a sort, and also are added, subtracted, and multiplied in a comparable way. (Division, which is additional down the page, is a bit different.) First, though, you"ll probably be inquiry to show that you recognize the definition of complicated numbers.

You are watching: Simplify the expression (3-i)-(2+6i)

Solve 3 – 4i = x + yi

Finding the answer to this requires nothing more than knowing that two facility numbers have the right to be equal just if their real and also imaginary components are equal. In various other words, 3 = x and also –4 = y.

To simplify complex-valued expressions, you combine "like" terms and also apply the various other methods you learned for working through polynomials.

Simplify (2 + 3i) + (1 – 6i).

(2 + 3i) + (1 – 6i) = (2 + 1) + (3i – 6i) = 3 + (–3i) = 3 – 3i

Simplify (5 – 2i) – (–4 – i).

(5 – 2i) – (–4 – i)

= (5 – 2i) – 1(–4 – i) = 5 – 2i – 1(–4) – 1(–i)

= 5 – 2i + 4 + i= (5 + 4) + (–2i + i)

= (9) + (–1i) = 9 – i

You may discover it advantageous to insert the "1" in front of the second set of bracket (highlighted in red above) so you can much better keep monitor of the "minus" gift multiplied through the parentheses.

Simplify (2 – i)(3 + 4i).

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(2 – i)(3 + 4i) = (2)(3) + (2)(4i) + (–i)(3) + (–i)(4i)

= 6 + 8i – 3i – 4i2 = 6 + 5i – 4(–1)

= 6 + 5i + 4 = 10 + 5i

For the last example above, FOILing works for this kind of multiplication, if you learned that method. However whatever method you use, remember the multiplying and including with complexes works just like multiplying and including polynomials, other than that, while x2 is just x2, i2 is –1.You have the right to use the exact same approaches for simplifying complex-number expressions together you carry out for polynomial expressions, but you deserve to simplify also further through complexes since i2reduces come the number –1.

Adding and also multiplying complexes isn"t too bad. It"s once you work with fractions (that is, v division) the things turn ugly. Most of the reason for this ugliness is actually arbitrary. Remember earlier in primary school school, when you an initial learned fractions? your teacher would acquire her panties in a wad if you offered "improper" fractions. For instance, friend couldn"t speak " 3/2 "; you had actually to convert it to "1 1/2". But now that you"re in algebra, nobody cares, and you"ve most likely noticed the "improper" fractions space often much more useful than "mixed" numbers. The problem with complicated numbers is the your professor will acquire his boxers in a bunch if you leave imaginaries in the denominator. So how do you manage this?

This is quite "simple", however they desire me to eliminate that i underneath, in the denominator. The 2 in the denominator is fine, but the ns has obtained to go. To do this, i will usage the fact that i2 = –1. If ns multiply the fraction, top and also bottom, through i, then the ns underneath will vanish in a puff the negativity:

This was basic enough, yet what if they offer you something much more complicated? Simplify

If ns multiply this fraction, top and also bottom, by i, I"ll get:

Since ns still have an i underneath, this didn"t assist much. So how do I take care of this simplification? I usage something dubbed "conjugates". The conjugate the a facility number a + bi is the very same number, yet with the opposite sign in the middle: a – bi. When you multiply conjugates, friend are, in effect, multiply to develop something in the sample of a difference of squares:

Note that the i"s disappeared, and the final result was a sum of squares. This is what the conjugate is for, and also here"s how it is used:

So the price is

In the last step, note how the fraction was break-up into 2 pieces. This is because, technically speaking, a complex number is in 2 parts, the real part and the ns part. Lock aren"t claimed to "share" the denominator. Come be sure your price is totally correct, break-up the complex-valued portion into that two separate terms.

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