How do I simplify this?
-5+i/2i
I have been having trouble with the imaginary number concept and would like some help with this question. Thanks so much!
Multiply the numerator and denominator by the conjugate of the denominator to find
2
+
i
1
−
i
=
1
2
+
3
2
i
Explanation:
The conjugate of a complex number
a
+
b
i
is
a
−
b
i
. The product of a complex number and its conjugate is a real number. We will use this fact to eliminate the complex number from the denominator of the given expression.
2
+
i
1
−
i
=
(
2
+
i
)
(
1
+
i
)
(
1
−
i
)
(
1
+
i
)
=
2
+
2
i
+
i
−
1
1
+
i
−
i
+
1
=
1
+
3
i
2
=
1
2
+
3
2
i
sorry the numbers got messed up
All good! Thank you so much! You are very helpful!!
since i^2 = -1,
1/i = -i
so (-5+i)/(2i) = (-5+i)(-i/2) = 1/2 + 5/2 i
or, using the conjugate,
(-5+i)/(0+2i) = (-5+i)(0-2i)/(0^2+2^2) = (-5+i)(-2i)/4 = 1/2 + 5/2 i
To simplify the expression -5 + i/2i, we need to rationalize the imaginary denominator. The original expression can be rewritten as:
-5 + i/(2i)
To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of 2i is -2i, so we have:
(-5 + i/(2i)) * (-2i/-2i)
Multiplying the numerator and denominator, we get:
(-5 * -2i) + (i * -2i) / ((2i) * (-2i))
Simplifying further:
(10i - 2i^2) / (-4i^2)
Now, we substitute i^2 with -1:
(10i - 2(-1)) / (-4(-1))
Simplifying further:
(10i + 2) / 4
Now, we can simplify the fraction by dividing both the numerator and denominator by 2:
10i/4 + 2/4
Simplifying:
(5i/2) + 1/2
So, the simplified expression is (5i/2) + 1/2