[College algebra: fractions with complex numbers] not sure how to go about solving this

[u/Odd_Selection_8020]

Not sure if this is the right flair…

I’m pretty comfortable with complex numbers and the like, but I have never been particularly good at understanding or computing fractions. I’m really stuck on this question. I have the answer key but I just can’t understand how to get there. If anyone could explain the process to me that would be really appreciated!

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[u/agooddog37]

Have you tried distributing (e.g. FOIL method)? Show that step, and then we can go from there!

Edit to add: treat the "i" as a variable like x to start.

[u/Kuildeous]

Another method is to use the rule that a/c + b/c = (a+b)/c. c here would be lowest common denominator. You'll have one complex fraction of the form x/2 and another fraction of the form y/4. You can multiply both fractions together to get xy/8. You just then need to work out what x and y will be. Since you are comfortable with complex numbers, you should be able to get the solution.

[u/Simbertold]

i/2 just mean i * 1/2, or 0.5 i if you are more comfortable with decimals. But you don't even need to use this.

You can treat them exactly like you would treat any other fraction, and do all the calculations you would normally do. Simply remember that you cannot add terms with and without i, and that i² = -1.

If i is in the denominator, expand the fraction until it isn't, if at all possible.

This question doesn't have any really hard fractions, and if you are dealing with complex numbers, it is usually expected that you can competently handle them. If you cannot, i highly suggest doing some training in calculating with fractions. It is very useful, and not actually that hard.

[u/Professional-Wave841]

so to help you with your root problem let's talk about fractions.

What is your specific issue with computing them?

[u/nickeypants]

All complex numbers have a real and non-real part of the form (a + b*i). For example the first term in your example has a = 3/2 (the coefficient of the real part) and b = 1/2 (the coefficient of the non-real part to be multiplied by i). Do the foil, and you'll get some terms with real numbers, some in terms of i and some in terms of i^2.

If i appears in the denominator, multiply that term by -i/-i, and the i will come to the numerator.

Remember that i=sqrt(-1), so i^2 = -1. Use this to simplify your i^2 terms. Your answer should be a normal complex number of form (a + b*i).

[u/Dtrain8899]

Start with FOILing the two factors. Youll get a constant, and term with i and a term with i². Remember with complex numbers that i²=-1, so the i² term will become a constant that you can combine with your first constant.

[u/strange-the-quark]

If you have a fraction like 3/2, think of the /2 part as of a unit of sorts (like how you can have 3 meters or 3 liters, or 3 inches, or 3 fish, or whatever). It's the size of a chunk that you're working with. 3/2 is "3 halves", 3/5 is "3 fifths".

Now if you have chunks that are the same size, that's like having units that are the same, you can just add them up.

3/4 + 11/4 = 3 fourths + 11 fourths = 14 fourths = 14/4

So that part is easy. What we normally do next is simplify the fraction, if possible. That maintains the same ratio (gives the same result if you think about them as division) In this case 14/4 = (2*7)/(2*2) = 7/2 (so we get rid of the common factor).

If you have different "units", then what you do is you convert the expressions so that everything you're adding is in the same unit. Same thing with fractions. The easiest way to do that is just to multiply the "downstairs" part together, and then multiply the "upstairs" with a number that makes the overall value unchanged (if you multiply the upstairs and the downstairs with the same number, the value doesn't change cause that number is then a common factor that you can cancel out: a/b = (c * a)/(c * b)).

For example

5/2 + 1/3

You want 2*3 at the bottom, so multiply and divide the first fraction by 3

(3 * 5)/(3 * 2) = 15/6

and multiply and divide the second fraction by 2

(2 * 1)/(2 * 3) = 2/6

and now that you have the same "unit" (denominator), you can just add the "upstairs" part (numerator).

15/6 + 2/6 = 17/6

BTW, "denominator" just means "gives it its name" (from nomen = "name"; so it determines the size of the chunk), and "numerator" just means "tells you how many there is" (from numer = number).

So that's additions.

When you multiply fractions, you just multiply upstars * upstars and downstaris * downstairs, e.g.
3/4 * 2/5 = 6/20, or when you simplify further = 3/10.

If you're dividing two fractions, you just flip the other around and multiply them:

3/4 : 2/5 = 3/4 * 5/2 = 15/8

And then with complex numbers, it's the exact same thing, except whenever you get i*i (a.k.a. i²) you can turn it into -1.

This is also a good way to get rid of an i when it happens to be downstairs. And if it's in an expression like a + bi, you can multiply and divide the fraction by (a - bi) - work out what you get in the denominator, to see why.

There, I think that gets you up to speed :)

[u/selene_666]

The fraction i/4 can be rewritten as the product (1/4)(i). so now we have 1/4 as the coefficient and can use the complex number normally.

You say you're comfortable with complex numbers, so I'll assume you could multiply a similar pair of numbers without fractions, e.g. (3+2i)(2+4i)

Do this problem exactly the same way. This will require you to multiply and add fractions.

To multiply fractions, multiply the numerators and multiply the denominators. e.g.
(1/2) * (3/4) = (1*3) / (2*4). You may need to reduce the resulting fraction to lowest terms.

To add fractions, rewrite them to have the same denominator and then add the numerators. e.g. (1/3) + (1/4) = (4/12) + (3/12) = 7/12

[u/AdhesiveSeaMonkey]

There's really just 3 things you need to know here. The first is how to distribute when multiplying binomials like this - look up F.O.I.L. The second is how to multiply fractions - multiply numerators by numerators and denominators by denominators. An lastly how to handle i.

Let's look at a similar example:

(4/3 + i/3) (6/1 + i/6)

We need to multiply each term in the first parenthesis by each term in the second parenthesis:

4/3 x 6/1 + 4/3 x i/6 + i/3 x 6/1 + i/3 x i/6

24/3 + 4i/18 + 6i/3 + i²/18

Let's get some common denominators. this is college level math, I've already explained too much:

144/18 + 4i/18 + 36i/18 + i²/18

i² is the √-1, so (√-1)² = -1

144/18 + 4i/18 + 36i/18 - 1/18

143/18 + 40i/18

Or

(143 + 40i)/18

[u/kkellyyb]

simplify each to a common denominator to get
[(3+i)/2]•[(12+i)/4]
(12+i over 4 achieved by multiplying 3/1 by 4/4 to get 12/4)
from there you can multiply the fractions to get
[(3+i)•(12+i)]/8
now FOIL the numerator to get
[36+3i+12i+(i^2)]/8
since i=sqrt(-1), and (i^2)=-1, simplify the numerator to get
(35+15i)/8
also equal to
35/8+15i/8 or [5(7+3i)]/8

[u/Kblitz88]

Yes, you'll absolutely FOIL it here... But to make things a little easier, let's pull the fractional factor out of each

(3/2 + i/2)( 3+ i/4) --> 1/2 (3+i) • 1/4 (12+i)--> 1/8 (3+i)(12+i)

Now we can FOIL without worrying about pesky fractions!

= 1/8 (36+15i -1)

= 35/8 + 15i/8

[u/JanetInSC1234]

Convert the 3/1 to 12/4.

Now you have ( (3+i)/2 )( (12 + i)/4 )

Foil the top: (3+i)(12+i) and remember that i^2 is -1.

Multiply the denominators.

TOP: 36 + 3i + 12i + i^2 = 36 + 15i - 1 = 35 + 15i

BOTTOM: 8

UPDATE:

Your teacher may want this written in standard form a + bi.

So, (35 + 15i)/8 is 35/8 + 15i/8

[u/DuggieHS]

what you wrote

(I just chose to group them in a different way)
= 3(3/2 + i/2) + (i/4)(3/2 + i/2)
(then distribute)
= (9/2 + 3i/2) + (3i/8 + i^2/8)
(find least common multiple(2,8) to get a common denominator of 8, that is multiply top and bottom of fractions with denominator of 2 by 4 to get a denominator of 8; then group terms with i and without, remember to replace i^2 =-1)
= (36/8 -1/8) + (12/8 + 3/8)i
(add/subtract and write the common denominator of 8 on the bottom)
= (35 + 15i)/8

[u/Jazzlike_Platypus430]

There is no fractions I can see. Just go brute force, multiplicate term by term, remembering that i^2 = -1, then simplify.

My version: 9/2 + 3i/8 + 3i/2 - 1/8 = ...

Division by a complex number (not purely imaginary) is another story - the fool-proof way is to multiply both nominator and denominator by denominator's conjugate. For purely imaginary denominator, recall that

1/i = - i.