What is the multiplicative inverse of 6-3i?
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The multiplicative inverse of an element of a set is another element of this set such that the product of the two elements is a multiplicative identity. In the given set of complex numbers, the multiplicative identity is 1. This is because multiplication by 1 does not change any complex number:
(a + bi)*1 = a + bi for any a and b.
Then, to find multiplicative inverse of 6 - 3i, we need to find the complex number such that the product of 6 - 3i and that number is 1. We can find it by performing division of 1 by 6 - 3i:
1/(6 - 3i) = ?
To find the result, multiply the numerator and denominator of the fraction above by the complex conjugate of 6 - 3i: 6 + 3i.
1/(6 - 3i) = (6+3i)/[(6-3i)(6+3i)]
In the denominator, the product of the two complex conjugates results in a real number:
(6-3i)(6+3i) = 36 - 9i^2 = 36 + 9 = 45. (Keep in mind that i^2 = -1.)
Therefore,
1/(6 - 3i) = (6 + 3i)/45 = 2/15 + (1/15)i.
The multiplicative inverse of 6 - 3i is 2/15 + (1/15)i.
This could be double-checked by performing the multiplication:
(6-3i)(2/15 + (1/15)i) = 12/15 - (6/15)i + (6/15)i - (3/15)i^2 = 4/5+ 1/5 = 1, as expected.
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The multiplicative inverse of a term x is given by y such that x*y = 1.
Now we have to find the multiplicative inverse of 6 - 3i
Let it be x + yi
(6 - 3i)(x + yi) = 1
=> x + yi = 1/(6 - 3i)
=> x + yi = (6 + 3i)/(6 - 3i)(6 + 3i)
=> x + yi = (6 + 3i)/ (6^2 + 3^2)
=> x + yi = (6+ 3i)( 36 + 9)
=> x + yi = (6 + 3i)/ 45
=> x + yi = 6/45 + (3/45)i
The multiplicative inverse is 6/45 + (3/45)i
The multiplicative inverse is the fraction whose numerator is 1 and denominator is the given complex number.
We'll note the inverse as x:
(6-3i)*x = 1
We'll divide by (6-3i) both sides:
x = 1/(6-3i)
Since it is not allowed to keep a complex number to denominator, we'll multiply the ratio by the conjugate of the complex number:
x = (6+3i)/(6-3i)(6+3i)
We'll re-write the denominator as a difference of squares:
(6-3i)(6+3i) = 6^2 - (3i)^2
(6-3i)(6+3i) = 36 - 9i^2
But i^2 = -1:
(6-3i)(6+3i) = 36 + 9
(6-3i)(6+3i) = 45
We'll re-write x:
x = (6+3i)/45
x = 6/45 + 3i/45
We'll simplify and we'll get:
x = 2/15 + i/15
The multiplicative inverse of the complex number 6 - 3i is 2/15 + i/15.
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