Using a+bi in different forms

[marshall1432]

Here are my set of problems I have:

1. What is the square root of -4?

my answer: None, because it can't be a negative number.

2. Write the following complex number in standard form, a + bi. Express numbers in fractional form if they are not integers: (1/2 - 2i) - (3/2 + i)

my answer: 2+3i

(added the 1/2 and 3/2 to get 2 and 2i and i to get 3i)

3. Write the following complex number in standard form, a + bi. Express numbers in fractional form if they are not integers:

-------
1-i^3

my answer: 3+3i

4. Write the following complex number in standard form, a + bi. Express numbers in fractional form if they are not integers: (1 + 2i)(4 - 3i)

my answer: 4+1i

5. Write the complex number (1 - 3i)/(2 - 2i) in the standard form a + bi, with a and b in fractional form.

my answer: 1/2a+3/2i

can anyone help me with these? thanks a lot!

 

[stapel]

1) Kind of the point of complex numbers is that you can take the square root of negative numbers. (Has your class somehow arrived at the complex-number homework without having mentioned imaginary numbers nor what "i" means?)

2) You might want to review how to work with negative numbers: 1/2 - 3/2 is not 4/2.

3) I will guess that "1 - i³" is the denominator of some fraction, but what is the numerator? How did you arrive at your answer?

4) How did you arrive at your answer? Please start with how you multiplied out ("FOILed") the product.

5) It almost looks like you said "(a + b)/(c + d) = a/c + b/d"...? Of course, you wouldn't do that, but it would be helpful if you showed your work, starting with what you used to rationalize the denominator.

Thank you.

Eliz.

 

[Mrspi]

Here are my set of problems i have:

1. What is the sqrt of -4?

my answer: None, bc it can't be a -<------Incorrect. Since you are apparently studying complex numbers, it IS possible to find sqrt(-4).

sqrt(-4) = sqrt(-1*4), or sqrt(-1)*sqrt(4). Recall that i = sqrt(-1), and that sqrt(4) = 2. So, sqrt(-4) = 2i


2. Write the following complex number in standard form, a + bi. Express numbers in fractional form if they are not integers.
(1/2 - 2i) - (3/2 + i)

my answer: 2+3i<-------incorrect. You are SUBTRACTING (3/2 + i), and subtraction is the same as adding the opposite. Rewrite the problem without parentheses, changing the sign of each term in the second set of parentheses:

1/2 - 2i - 3/2 - i
Now, combine like terms......
(added the 1/2 and 3/2 to get 2 and 2i and i to get 3i)

3. Write the following complex number in standard form, a + bi. Express numbers in fractional form if they are not integers:

-------
1-i^3

my answer: 3+3i<-------incorrect, if your problem was 1 - i^3 (I'm confused by the "bar" over the expression.

IF it is 1 - i^3, then you need to think about what i^3 is.
i^1 = i
i^2 = -1
i^3 = i^2 * i, or -1*i, or -i

So, 1 - i^3 becomes 1 - (-i) or 1 + i


4. Write the following complex number in standard form, a + bi. Express numbers in fractional form if they are not integers.
(1 + 2i)(4 - 3i)

my answer: 4+1i<-------Incorrect.

Multiply the two binomials together:
(1 + 2i)(4 - 3i) = 1*4 + 1*(-3i) + 2i*4 + 2i*(-3i)

4 - 3i + 8i - 6i^2
But i^2 = -1, so we can substitute -1 for i^2:
4 - 3i + 8i - 6(-1)
4 - 3i + 8i + 6

NOW combine like terms.


5. Write the complex number (1 - 3i)/(2 - 2i) in the standard form a + bi, with a and b in fractional form.

my answer: 1/2a+3/2i<------Incorrect. You don't want to end up with i in the denominator (because it is a radical, remember?)

Multiply numerator and denominator of the fraction by (2 + 2i):

(1 - 3i)(2 + 2i)
----------------
(2 - 2i)(2 + 2i)

Do the multiplications in numerator and denominator, as illustrated in the previous problem:

1 + 2i - 6i - 6i^2
----------------
4 + 4i - 4i - 4i^2

Substitute -1 for i^2.....
1 + 2i - 6i + 6
----------------
4 + 4i - 4i + 4

Combine like terms:

7 - 4i
------
8

Separate into real and imaginary parts to get the answer in a + bi form:

(7/8) - (1/2)i



can anyone help me with these? thanks a lot!

Looks to me like you need to do some intensive work to understand complex numbers!

 

[marshall1432]

thank you so much for the help. i worked them all out the way you showed me and here is what i got:

1. 2i
2. 2+i (not sure)
3. the bar is a fraction bar. The problem i was given had nothing as the numerator but did in fact have a denominator of 1-i^3
4. 5i+2
5. (7/8) - (1/2)i (not sure)

 

[stapel]

i worked them all out the way you showed me and here is what i got:

1. 2i

I'm not sure what you mean by "working this out", since you have shown no work and you were given this answer...?

2. 2+i (not sure)

As was explained earlier, no, 1/2 - 3/2 does not equal 1/2 + 3/2 = 4/2 = 2. So this would still be incorrect. (Since we cannot re-teach the background material on integers, arithmetic with negatives, and fractions, you might want to hire a tutor and spend a week or two reviewing these topics.)

3. the bar is a fraction bar. The problem i was given had nothing as the numerator but did in fact have a denominator of 1-i^3

If you were given a fraction with no numerator, then there is no way to proceed. (If you'd been given the complex conjugate, you would of course have recognized this notation from your class and your book as not being a "fraction", so I will trust you that this actually is a fraction.)

Please ask your instructor for the rest of the fraction.

4. 5i+2

No: 4 + 6 does not equal 2. Please review the nearly-complete solution you were provided, and re-work that one last step.

5. (7/8) - (1/2)i (not sure)

Why are you "not sure" of the complete worked solution the tutor provided you? Please clarify. Thank you.

Eliz.

 

[marshall1432]

think i got it

1. 2i
2. -1+3i
3. i'll ask about the bar
4. 10+5i
5. i just wasnt sure if it was in the right form or not.

 

[stapel]

2. -1+3i

It's unfortunate that you still won't show us your work. If, at some point, you could show what you are doing, we might have better luck in figuring out where you're going wrong. My current guess is that you still need to follow the recommendation, made various times in the past, to learn how to work with negative numbers. But I can't be sure...?

My best wishes to you.

Eliz.

 

[skeeter]

Re: think i got it

1. 2i correct

2. -1+3i should be (-1 - 3i)

3. i'll ask about the bar the "bar" notation above the the complex number indicates the conjugate of 1 - i³ = 1 + i ... which is 1 - i.

4. 10+5i correct

5. (1-3i)/(2-2i)*(2+2i)/(2+2i) = (8-4i)/8 = 1 - i/2