Having the following complex numbers A=5-2i and B=3+8i a) A+B b) B-A c)

Sapewa

Sapewa

Answered question

2021-12-28

Having the following complex numbers A=52i and B=3+8i
a) A+B
b) BA
c) A×B
d) What is the complex conjugate of A and the complex conjugate of B?
e) BA (B divide by A)

Answer & Explanation

Durst37

Durst37

Beginner2021-12-29Added 37 answers

Step 1
a) Given complex numbers
A=52i
B=3+8i
In addition of complex numbers real terms are added and imaginary terms are added together.
(a+ib)+(c+id)=(a+c)+i(b+d)
Step 2
so the sum of given numbers
A+B=(52i)+(3+8i)
=(5+3)+i(2+8)
=8+6i
Step 3
b) In subtraction of complex numbers real terms are subtracted and imaginary terms are subtracted together.
(a+ib)(c+id)=(ac)+i(bd)
Step 4
so the difference of the terms
BA=(3+8i)(52i)
=(35)+i(8+2)
=2+10i
Step 5
c)The product of two complex numbers can be found by
(a+ib)×(c+id)=(acbd)+i(ad+bc)
Step 6
Here the product is
A×B=(52i)×(3+8i)
=(5×3(2)×8)+i(8×5+(2)×3)
=31+34i
Step 7
d)The complex conjugate of
a+ib is aib
aib is a+ib
Step 8
The complex conjugate of
A=52i
A=5+2i
B=3+8i
B=38i
Step 9
e) In complex numbers for division multiply the numerator and denominator by conjugate of denominator
a+ibc+id=a+ibc+id×cidcid
=(acbd)+i(ad+bc)c2+d2
Step 10
Here the division of B by A will give
BA=(3+8i)(5+2i)(52i)(5+2i)
=(1516)+i(40+6)52+2
peterpan7117i

peterpan7117i

Beginner2021-12-30Added 39 answers

Step 1
a) (52i)+(3+8i)
=(5+3)+(2+8)i
5+3=8
2+8=6
=8+6i
Step 2
b) (3+8i)(52i)
=(35)+(8+2)i
35=2
8+2=10
=2+10i
Step 3
c) (52i)×(3+8i)
=(5×3(2)×8)+(5×8+(2)×3)i
5×3(2)×8=31
5×8+(2)×3=34
=31+34i
Step 4
e) 52i3+8i
Apply complex arithmetric rule:
a+bic+di=(cdi)(a+bi)(cdi)(c+di)=(ac+bd)+(bcad)ic2+a2
a=5, b=2, c=3, d=8
=(5×3+(2)×8)+(2×35×8)i32+82
Refine
=146i73
Rewrite 146i73 in standard complex form: 1734673i
=1734673i
Vasquez

Vasquez

Expert2022-01-08Added 597 answers

Step 1
a) Your input: simplify and calculate different forms of 3+52i+8i
Rewrite: 3+52i+8i=8+6i
The expression is already simplified.
Step 2
b) 3(52i)+8i
Rewrite: 3(52i)+8i=2+10i
The expression is already simplified.
Step 3
c) (3+8i)×(52i)
Use FOIL to multiply (for steps, see foil calculator), don't forget that i2=1
((3+8i)×(52i))=(31+34i)
Hence, (3+8i)×(52i)=31+34i
Step 4
e) 52i3+8i
Rewrite: 52i3+8i=(38i)(52i)73
Multiply using the formula a(b+c)=ab+ac
38i73=(173)(3)+(173)(8i)=3738i73
So,
(38i73)(52i)=(3738i73)(52i)
Use FOIL to multiply (for steps, see foil calculator), don't forget that i2=1
((3738i73)(52i))=(17346i73)
Hence, (38i)(52i)73=17346i73

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