for the division of complex numbers the numerator and the denominator must be multiplied by the cojugate denominator (the opposite sign)
(5) 3/(2i) the conjugate denominator is -2i = (3*-2i)/(2i)(-2i) = -6i/-4i = (-6/-4)i = 1.5i (am not sure can ant1 confirm this please)
(6) 3/(2 + i) = 3(2-i)/(2+i)(2-i) numerator: 6-i denomiator: this is of the form (a+b)(a-b) = a^2-b^2 = 4-i^2 i ink i and j are the same so i^2= -1 therefoore 4-i^2= 4+1=5
SANTOSH ....as for question 1 i think it is the easiest because it has no calculating involved....once you can see what exactly they are looking for then you know how to look at the question.....try and get into the mind of the teacher and see what they are trying to make you see......therefore for QUESTION 1 which is
3-4i = x + yi
ent if you put a bracket around -4i then the equation becomes
3 + ( -4i )? because if you expand the bracket a + multiple by a - equals a - therefore there is no change in the equation except it makes it easy to see for the next part..
when you put the equations like this
3 + (-4i) = x + yi
because both equations are equal isn't x = 3 and y = -4
hey guys i was reading through and i was jus wondering...for the first comment when Santosh answered No.5 ...Simplify 3/2i...i think there is an error...correct me if i'm wrong but remember when you multiply by the conjugate...we will get [3 x (-2i)]/ [2i x (-2i)] which will work out to be (-6i)/(-4i^2) so since the denominator has 'i' squared we can multiply it by (-1) which will give (-6i)/(4) which simplifies to (-3/2 i) or -1.5i...the difference from his answer and mine is the sign...
Rememeber that this is multiplication and you need to multiply each term in the bracket by the other term in the other bracket. 2 * 3 and 2 * 4i and -i * 3 and - i * 4i 6 + 8i - 3 - 4i
well swanky thing the conjugate for quse. 5 is -2i since the no. is positve and for ques. 6 the conjugate is 2-1 since it was positive. so the quetsionwill then look like
QUESTION 1 3 – 4i = x + yi as we notice the commons and non common.what do we see that has similarity. The 3 and x and 4i and yi x is proportional to 3 yi is proportional to 4i x=3, yi=4i or y=4
QUESTION 5 3/(2i) Firstly we know we need conjunction in order to slove which is -2i. denorminator= (2i)(-2i) = -4i^2 remembering i^2 is equal to -1 so =-4(-1)=4 and for numerator 3(-2i) =-6i which will equal to -6i/4
QUESTION 6 3/(2 + i)
First things first we multiply the denominator and the numerator by the conjugate of the of the denominator which is +2
= 3(2 - i)/ (2 + i x 2 - i)
(a+b)(a-b) = a^2 - b^2 cal.for denominator
(2 + i)(2 - i) = 2^2 - i^2 = 4 - (-1) = 1+1 = 5 (denorminator) and 3(2 - i) = 6 - 3i (numerator) which will be equal to = 6 - 3i/ 5
5. 3/2i When dividing we multiply by the conjugate. The conjugate of 2i is -2i 3/2i * -2i/-2i [2i * -2i] gives us 2^2 – i ^2 4 - (1) = 3 Then we say 1/3 multiply by (3) (-2i)
Diesel wanted to find out how to divide one.Well,in the following question: 4 + 3j/(2+5j) Step 1:Multiply by the conjugate of (2+5j) and when is addition it is minus,therefore it would be (2-5j)
Secondly,find the difference in squares (a+b)(a-b)=a^2 +b^2 Solving the denominator= (2+5j) (2-5j) =2^2-(5j)^2 =4-25(-1) =4+25=29
For the num.(4+2j)(2-5j) we multiple and always remember to join our arrows from one to another and multiple them one at a time. Therefore,we will get : 8-20j+6j-15j^2 =23-14j
#6 3/(2 + i) u multiply by the conjugate {3/(2+i)}* {(2-i)/(2-i) multiply out the denominator 2^2 - i^2 [i^2 = -1] 4 - -1 [4 + 1] 5 now u multiply out the numerator 3/5 * (2 - i) and u'll get the ans.
I would say that since a equal sign is present indicating that what is on the right hand side must be equal to the left hand side so you would get x to be 3 and y to be 4i
Why not allow our creator, Jesus, to show His love? Make Him your best friend by telling him the things you tell your best friend. When Jesus is part of your everyday life, well you are on top of the world. The best things will happen and the trials will be testing but the best for you will prevail.
(2) 2 + 3i +
ReplyDelete1 – 6i
= 3 - 3i
(3) 5 – 2i –
–4 – i
= 9 - i
for the division of complex numbers the numerator and the denominator must be multiplied by the cojugate denominator (the opposite sign)
(5) 3/(2i)
the conjugate denominator is -2i
= (3*-2i)/(2i)(-2i)
= -6i/-4i
= (-6/-4)i
= 1.5i (am not sure can ant1 confirm this please)
(6) 3/(2 + i)
= 3(2-i)/(2+i)(2-i)
numerator: 6-i
denomiator: this is of the form
(a+b)(a-b) = a^2-b^2
= 4-i^2
i ink i and j are the same so i^2= -1
therefoore 4-i^2= 4+1=5
= 6-i/5
=6/5 - i
can any1 help with (1)
2. Simplify (2 + 3i) + (1 – 6i)
ReplyDelete2 + 3i
+
1 – 6i
_________
3 – 3i
2. Simplify (5 – 2i) – (–4 – i)
ReplyDelete5 – 2i
-
-4 – i
_______
9 - 1i
2. Simplify (2 – i)(3 + 4i)
ReplyDeleteMultiply by looping terms in both brackets
First term in first bracket by the two terms in the second bracket.
Second term in the first bracket by the two terms in the second brackets.
6 + 8i - 3i - 4i2
Since i^2 = -1
6 + 5i -4(-1)
6 + 4 +5i
=10 + 5i
when u haer the word Simplify or they sau to Simplify in a questions this means that have to break in down or work it out into its Simplest terms
ReplyDeleteExpress in terms of j: √-2√-18
ReplyDeletecan some one help me with this...
santosh i think your answer that u got for part 6 is wrong because when u worked out the numerator u got (6 -i) but the answer is 6 - 3i
ReplyDeletebeacause when multipling u multiple but terms your denomiator is correct (5)
so the final answer is 6 - 3i / 5
which can be broken down into 6/5 - 3i/5
if you want u can work it out in your calualator but i beleive that it should be ok to leave it like that
ok dark angel what they are saying is just to simplify in terms of j
ReplyDelete=√-2√-18
= √-1 √2 X √-1 √8
= √i^2 √2 X √i^2 √8
= (i)(√2) X (i)(√8)
then u can multiple the terms to get
= (i^2)(√16) but i^2=-1
so = -1 X 4
= -4
did this help you dark angel
#6. 3/(2 + i)
ReplyDeleteThe firs thing to do is to multiply the denominator and the numerator by the conjugate of the of the denominator.
conjugate of 2 + i => 2
Therefore you get: 3(2 - i)/ (2 + i x 2 - i)
working out denominator: (recall - (a+b)(a-b) = a^2 - b^2
(2 + i) (2 - i) = 2^2 - i^2
= 4 - (-1) = 1+1 = 5
working out the numerator:
3(2 - i) = 6 - 3i
Therefore:
6 - 3i/ 5
Yes renjisan i understand... but i didnt ask to simplify i wanted the answer in terms of j...
ReplyDeleteSANTOSH ....as for question 1 i think it is the easiest because it has no calculating involved....once you can see what exactly they are looking for then you know how to look at the question.....try and get into the mind of the teacher and see what they are trying to make you see......therefore for QUESTION 1 which is
ReplyDelete3-4i = x + yi
ent if you put a bracket around -4i then the equation becomes
3 + ( -4i )? because if you expand the bracket a + multiple by a - equals a - therefore there is no change in the equation except it makes it easy to see for the next part..
when you put the equations like this
3 + (-4i) =
x + yi
because both equations are equal isn't
x = 3 and y = -4
can you see it now ?
hey renjisan, thanks. i cant believ i make such silly mistakes
ReplyDeletei diden't even realise that jhonsmith, thanks
ReplyDeleteFor number 1, The first step i think is to group like terms together, real with real and imaginery with imaginery. Then look out for the police!
ReplyDeleteFor number 2, the strategy that I would use is:
Real Imaginery
2 + 3i
+
1 - 6j
Then, solve.
For number 3, it’s the same strategy as number 2 but just equiped with a lot more police.
Number 4 can be recognised as multiplication on the spot. Firstly, I’ll draw loops to guide me so that I don’t skip or miss out any term
Then I’ll simplify, add like terms while watching out for police and then simplify.
1) 3 – 4i = x + yi
ReplyDeletex=3, y=4
2) (2 + 3i) + (1 – 6i)
= (3+3i)
3) (5 – 2i) – (–4 – i)
5- -4= 9 -2--1=1
=(9-1j)
4) (2 – i)(3 + 4i)
( 6 + 8i-3i-4i^2)
(6+5i+4)
(10 +5i)
5) 3/(2i) find the con. which is (-2i) work out the denom and then the num.
same thing applies for no6.
3 – 4i = x + yi
ReplyDelete3 – 4i is already in the form of x + yi so therefore it only to properly state what x and y is.
where x = 3
y = –4
hey guys i was reading through and i was jus wondering...for the first comment when Santosh answered No.5 ...Simplify 3/2i...i think there is an error...correct me if i'm wrong but remember when you multiply by the conjugate...we will get
ReplyDelete[3 x (-2i)]/ [2i x (-2i)] which will work out to be (-6i)/(-4i^2) so since the denominator has 'i' squared we can multiply it by (-1) which will give (-6i)/(4) which simplifies to
(-3/2 i) or -1.5i...the difference from his answer and mine is the sign...
no 2
ReplyDeletework out by adding common terms. look out for police
(2+3i) + (1-6i)= (3 - 3i)
no 3
ReplyDeletein this problem we have to pay special attention to police
(5-2i) - (-4-i)= 9-1i
no 4
ReplyDeletework out by multiplying each term in 1st bracket by all terms in 2nd bracket
(2-i)(3+4i)= 6+8i-3-4i
= 3+4i
no 5 and no 6 can be worked out by multiplying by the conjugate.
ReplyDeletefor no 5 the conjugate is -2i
for no 6 the conjugate is 2-i
Solve 3 – 4i = x + yi
ReplyDeleteAll you need to say is x = 3 and y = - 4
Simplify (2 + 3i) + (1 – 6i)
ReplyDeleteAll you have to say is numbers alone is real and anything with a letter is imaginery numbers
and you have to add the real and then add the imaginery
(2 + 1 ) ( 3 i - 6 i)
3 -3i
Simplify (5 – 2i) – (–4 – i)
ReplyDeleteLook out for your signs
9 - 1i
(2 – i)(3 + 4i)
ReplyDeleteRememeber that this is multiplication and you need to multiply each term in the bracket by the other term in the other bracket.
2 * 3 and 2 * 4i and -i * 3 and - i * 4i
6 + 8i - 3 - 4i
3 + 4i
For question 2:
ReplyDelete2 + 3i
+
1 – 6i
=
3 – 3i
for question 4
ReplyDelete(2 – i)(3 + 4i) in order to simplify, we expand the brackets which gives as follows
6 + 8i - 3i - 4i2
Since i squared is = -1
6 + 5i -4(-1)
6 + 4 +5i
=10 + 5i.
For question 3 look out for all the negative signs, remebre thata a negative multiplied by another negative will give a positive
ReplyDeleteCan someone help me with questiom 5 and 6 plez.
ReplyDelete2. (2+3i) + (1-6i)
ReplyDeleteu add all real nos. first
= 2+3
=5
then u add all imag.
=3+(-6)
=-3
so answer = 5-3i
sorry i aws watchin d wrong question
ReplyDeleteanswer= 3-3i
question 4
ReplyDelete(2-i)(3+4i)now this is a bit tricky if u're not careful so this is how to approach it.
open out the brackets and u would get
6+8i-3i+4
this happens bcuz u have a -1 and when u multiplt it the sign changes.
therefor u would end up with 10+5i
well swanky thing when u have questions like these u have to use the conjugate.
ReplyDeletei.e if it is positve the conjugate would be negative and vice versa. glad to help hope u can attempt the question now.
Explain to me whats the conjugate plez....
ReplyDeletequestion 1
ReplyDeletesince it is an eqaution waht u have on one side is equal to that on the other side therefore x=3 and y=+4
well swanky thing the conjugate for quse. 5 is -2i since the no. is positve and for ques. 6 the conjugate is 2-1 since it was positive. so the quetsionwill then look like
ReplyDelete3/2+1 (2-1/2-1)
Q2
ReplyDelete(2+3j)+ (1-6j)
=(2+1)+(3j-6j)
=3-3j
Q3
ReplyDelete(5-2j)-(-4-j)
=(5-(-4))-(-2j-j)
=9-j
Q6
ReplyDelete3/2+i
=3/(2+i)*(2-i/2-i)
denominator=(2+i)(2-i)= 4-2i+2i-i^2=4-i^2
when i^2= -1 =4-(-1)= 5
and the numerator= 3(2-i)= 6-3i
=(6-3i)/5
= 6/5 -3/5i
(5) Simplify 3/(2i)
ReplyDeleteWhen dividing complex nos. both the numerator and the denominator are multiplied by the conjugate of the denominator. In this case it is -2i
Numerator: 3(-2i)
=-6i
Denominator: (2i)(-2i)
= -4i^2
Since i^2 = -1, we multiply the denominator by -1: (-4)(-1)= 4
Therefore the ans is: -6i/4
= (-3/2)i
you are correct chun li
ReplyDeleteQ3
ReplyDelete(2-i)(3+4i)
=([2*3]+[2*4i]+[-i*3]+[-i*4i])
=6+8i+-3i+-4i^2
=6+5i-4(-1)
=10+5i
QUESTION 1
ReplyDelete3 – 4i = x + yi as we notice the commons and non common.what do we see that has similarity.
The 3 and x and 4i and yi
x is proportional to 3
yi is proportional to 4i
x=3, yi=4i or y=4
QUESTION 2
simple addition
(2 + 3i) + (1 – 6i) (+ + -=-)
= (3-9i)
QUESTION 3
also simple substraction
(5 – 2i) – (–4 – i)
5 - -4 = 9 (- + -=+)
-2 - -1 = 1 (signs cancel)
=(9-1j)
QUESTION 4
(2 – i)(3 + 4i)
(2X3 + 2X4 -3i -4i^2)
= (6 + 8i-3i-4i^2) as we remember i^2=-1
= (6+5i-4(-1))
= (6+5i+4)
= (10+5i)
QUESTION 5
3/(2i)
Firstly we know we need conjunction in order to slove which is -2i.
denorminator= (2i)(-2i)
= -4i^2 remembering i^2 is equal to -1 so
=-4(-1)=4
and for numerator 3(-2i)
=-6i
which will equal to -6i/4
QUESTION 6
3/(2 + i)
First things first we multiply the denominator and the numerator by the conjugate of the of the denominator which is +2
= 3(2 - i)/ (2 + i x 2 - i)
(a+b)(a-b)
= a^2 - b^2 cal.for denominator
(2 + i)(2 - i)
= 2^2 - i^2
= 4 - (-1)
= 1+1
= 5 (denorminator)
and
3(2 - i)
= 6 - 3i (numerator)
which will be equal to
= 6 - 3i/ 5
(2 + 3i)+(1 - 6i)
ReplyDeletesince this is addition we add real numbers together followed by imaginary numbers giving us
2 +1 = 3And
3i + (-6i) = -3i
(5 – 2i) – (–4 – i)
ReplyDeletefor subtraction we treat real numbers as usual i.e. [5 - (-4)] = 9
and imaginary
[-2i – (-i )]which is -2i + i = -i
(2 - i) (3 + 4i)
Multiplication: Make loops (2 * 3) - (i * 3) + (2 * 4i) - (-i * 4i)
6 - 3i + 8i + 4i^2
6 + 5i + 4(1)
6 + 5i + 4
10 + 5i
5. 3/2i
When dividing we multiply by the conjugate.
The conjugate of 2i is -2i
3/2i * -2i/-2i
[2i * -2i] gives us 2^2 – i ^2
4 - (1) = 3
Then we say 1/3 multiply by (3) (-2i)
Solve 3 – 4i = x + yi
ReplyDeletewhere x = 3
and y = -4
Simplify (2 + 3i) + (1 – 6i)
ReplyDelete2 + 3i
+
1 – 6i
= 3 - 3i
Simplify (5 – 2i) – (–4 – i)
ReplyDeletemaking it more simpler to understand
5 – 2i
-
–4 – i
= 9 - 1i
whatis an imaginery number?
ReplyDeleteImaginery meaning not real.
ReplyDelete2/ 2+3i
ReplyDelete+ 1-6i
= 3-3i
3/ 5-2i
--4-i
= 1-3i
4/ (2-i)(3+4i)
6+8i-3i-4i^2
6+4+8i-3i
= 10+5i
yes people can u do numbers 1,5 and 6 plezzzzzzzzzz!!!!!!!!!!
ReplyDeletei kno numbers 5 and 6 is with the conjugate thingny but still do it
Simplify 3/(2 + i)
ReplyDeleteu find d conj which is (2 - i)/(2 - i) thats d opp of the denominator.
therefore ur eq is 3/(2 + i) * (2 - i)/(2 - i)
work out the denominators = (2 + i)/(2 - i)
= 2^2 - i^2 = 4+1 = 5
so then u work out the numer... 3 * ( 2-i)
= (6 - 3i)/ 5
dont forget to put in yr denominator that u worked out 1st.. better now SMILEY???? lol
Simplify (2 + 3i) + (1 – 6i)
ReplyDelete2 + 3i
+
1 – 6i
= 3 - 3i
2/ 2+3i
ReplyDelete+ 1-6i
= 3-3i
3/ 5-2i
--4-i
= 1-3i
4/ (2-i)(3+4i)
6+8i-3i-4i^2
6+4+8i-3i
= 10+5i
2](2 + 3i) + (1 – 6i)
ReplyDelete2+3
1-6
=3-3j
please explain to me how to do the division ones
Diesel wanted to find out how to divide one.Well,in the following question:
ReplyDelete4 + 3j/(2+5j)
Step 1:Multiply by the conjugate of (2+5j) and when is addition it is minus,therefore it would be (2-5j)
Secondly,find the difference in squares
(a+b)(a-b)=a^2 +b^2
Solving the denominator= (2+5j) (2-5j)
=2^2-(5j)^2
=4-25(-1)
=4+25=29
For the num.(4+2j)(2-5j)
we multiple and always remember to join our arrows from one to another and multiple them one at a time.
Therefore,we will get :
8-20j+6j-15j^2
=23-14j
The final ans. would be with num/den
23-14j/29
ques2. (2 + 3i) + (1 – 6i)
ReplyDelete2+1=3
3i-6i=-3i
= 3-3i
Ques1.3 – 4i = x + yi
ReplyDeletesince the equation would be equal on both sides
x=3 and y=-4
Ques3.(5 – 2i) – (–4 – i)
ReplyDelete5-(-4)=5+4=9
-2i-i=-3i
=9-3i
This comment has been removed by the author.
ReplyDeleteQues3.in this question would the answer to -2i -(-i)be -2i+i which would be -i ir would it be -2i-i which would be -3i
ReplyDeleteQues3. (2-i)(3+4i)
ReplyDeletewell we need to open out the brackets
6+8i-3i-4i^2
6+8i-3i-4(-1)
6+4+8i-3i
= 10+5i
Ques3/(2i)or i now know how to work out this
ReplyDeleteconjugate thing...
question #3
ReplyDelete(5 – 2i) – (–4 – i)
subtract real from real and imaginary from imaginary;
(5 - -4)- (-2i - -i)
9 - (-2i + i)
9 - i
#6
ReplyDelete3/(2 + i)
u multiply by the conjugate
{3/(2+i)}* {(2-i)/(2-i)
multiply out the denominator
2^2 - i^2 [i^2 = -1]
4 - -1 [4 + 1]
5
now u multiply out the numerator
3/5 * (2 - i)
and u'll get the ans.
3 – 4i = x + yi
ReplyDeleteI would say that since a equal sign is present indicating that what is on the right hand side must be equal to the left hand side so you would get x to be 3 and y to be 4i
questions...answers
ReplyDelete√-1=j
j^2=-1
1.√-7
..√-1
..√7j
2.√-2√-8
..√-1√2,√-1√8
..j^2√16
3.√(-2)(-18)
..√(2j)(18j)
..√36j
√-1=j
ReplyDeletej^2=-1
4)-√-2/5
-√1√2/5
-j√2/5
5)j^2-j^6
-1-(-1)*(-1)*(-1)
6)(√-2)^2+j4
√-1√2√-1√2
j^2√4
-2
√-1=j
ReplyDeletej^2=-1
7)(6+7j)(3-5j)
(6+3)(7j-5j)
=9+2j
8)(12+6j)-(4+5j)
(12-4)(6j-5j)
=8+j
1.3-4i=x+yi
ReplyDelete2.[2+3i]+[1-6]
[2+1]+[1+6]
3-3i
3.[7+12j]+[-11-6j]
[7-11]+[12j-16j]
[-4-4j]
4.[-17j+4]-[18-3j]
[-17j-3j]-[4-18]
[-20j-14]
5.[3-11j]*[7+6j]
21+18j-77j-66
21-59j+66
45-59j
6.[15j-8]*[7+6j]
105j+90j^2-56-48j
105j-48j-56+90j^2
57j-146
7.15j+3/4+3j
15j+3/4+3j *4-3j/4-3j
60j-45j^2+12-9j
-----------------
[4+3j] [4-3j]=16-9j^2=25
=59j+45+12=59j+57=[59j 57]
1/25[59j;57]
MY OWN QUESTIONS
Complex`no`question
ReplyDelete5/3-√2j
3+√2j/3+√2j
3^2-√2j^2
9-2j^2
9+2
11
Complex`no`question
ReplyDelete(5 − √(-16)) × (2 + √(-9))
(5-√16j)x(2+√9j)
(5- 4j)x(2+3j)
(5 − 4j) × (2 + 3j)
10 + 15j - 8j - 12j^2
10 + 7j - 12 x -1
10 + 7j + 12
22 + 7j
Complex`no`question
ReplyDeletej(3 + 2j)(6 -4j)
(3j + 2j^2) (6 -4j)
18j + (-12j^2) + 12j^2 - 8j^3
18j - 8j^3
10j^2
10(-1)
-10
Complex`no`question
ReplyDeletej^6 + j^ 4 - j^2
j^2j^2j^2 + j^2j^2 - j^2
(-1)(-1)(-1) + (-1)(-1) - (-1)
(-1) + 1 - (-1)
0 - (-1)
= 1
2. Simplify (2 + 3i) + (1 – 6i)
ReplyDelete2 + 3i
+
1 – 6i
_________
3 – 3i