Having the following complex numbers A=5−2i and B=3+8i a) A+B b) B−A c) A×B d) What is the complex conjugate of A and the complex conjugate of B? e) BA (B divide by A)

Question

Having the following complex numbers A=5−2i and B=3+8i  a) A+B  b) B−A  c) A×B  d) What is the complex conjugate of A and the complex conjugate of B?  e) BA (B divide by A)

Durst37 · Accepted Answer

Step 1  a) Given complex numbers  A=5−2i  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=(5−2i)+(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)=(a−c)+i(b−d)  Step 4  so the difference of the terms   B−A=(3+8i)−(5−2i)  =(3−5)+i(8+2)  =−2+10i  Step 5  c)The product of two complex numbers can be found by  (a+ib)×(c+id)=(ac−bd)+i(ad+bc)  Step 6  Here the product is  A×B=(5−2i)×(3+8i)  =(5×3−(−2)×8)+i(8×5+(−2)×3)  =31+34i  Step 7  d)The complex conjugate of   a+ib is a−ib  a−ib is a+ib  Step 8  The complex conjugate of  A=5−2i  ⇒A―=5+2i  B=3+8i  ⇒B―=3−8i  Step 9  e) In complex numbers for division multiply the numerator and denominator by conjugate of denominator  a+ibc+id=a+ibc+id×c−idc−id  =(ac−bd)+i(ad+bc)c2+d2  Step 10  Here the division of B by A will give  BA=(3+8i)(5+2i)(5−2i)(5+2i)  =(15−16)+i(40+6)52+2

peterpan7117i · Accepted Answer

Step 1  a) (5−2i)+(3+8i)  =(5+3)+(−2+8)i  5+3=8  −2+8=6  =8+6i  Step 2  b) (3+8i)−(5−2i)  =(3−5)+(8+2)i  3−5=−2  8+2=10  =−2+10i  Step 3  c) (5−2i)×(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) 5−2i3+8i  Apply complex arithmetric rule:  a+bic+di=(c−di)(a+bi)(c−di)(c+di)=(ac+bd)+(bc−ad)ic2+a2  a=5, b=−2, c=3, d=8  =(5×3+(−2)×8)+(−2×3−5×8)i32+82  Refine  =−1−46i73  Rewrite −1−46i73 in standard complex form: −173−4673i  =−173−4673i

Vasquez · Accepted Answer

Step 1 a) Your input: simplify and calculate different forms of 3+5−2i+8i Rewrite: 3+5−2i+8i=8+6i The expression is already simplified. Step 2 b) 3−(5−2i)+8i Rewrite: 3−(5−2i)+8i=−2+10i The expression is already simplified. Step 3 c) (3+8i)×(5−2i) Use FOIL to multiply (for steps, see foil calculator), don&#039;t forget that i2=−1 ((3+8i)×(5−2i))=(31+34i) Hence, (3+8i)×(5−2i)=31+34i Step 4 e) 5−2i3+8i Rewrite: 5−2i3+8i=(3−8i)(5−2i)73 Multiply using the formula a(b+c)=ab+ac 3−8i73=(173)(3)+(173)(−8i)=373−8i73 So, (3−8i73)(5−2i)=(373−8i73)(5−2i) Use FOIL to multiply (for steps, see foil calculator), don&#039;t forget that i2=−1 ((373−8i73)(5−2i))=(−173−46i73) Hence, (3−8i)(5−2i)73=−173−46i73

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

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