(1 point) For some practice working with complex numbers: Calculate (4 + 6i) + (4 - 3i) = 8 + 3i, (4 + 6i) - (4 - 3i) = 9i, (4 + 6i)(4 - 3i) = 34+12i The complex conjugate of (1 + i) is (1 - i). In general to obtain the complex conjugate reverse the sign of the imaginary part. (Geometrically this corresponds to finding the "mirror image" point in the complex plane by reflecting through the x-axis. The complex conjugate of a complex number z is written with a bar over it: ? and read as "z bar". Notice that if z = a + ib, then (z) (?) = |z|^2 = a^2 + b^2 which is also the square of the distance of the point z from the origin. (Plot z as a point in the "complex" plane in order to see this.) If z = 4 + 6i then ? = ? and |z| = 7.2111. You can use this to simplify complex fractions. Multiply the numerator and denominator by the complex conjugate of the denominator to make the denominator real.
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The complex conjugate is $(1 - i)$. Next, we need to find the product of the given complex number and its conjugate: $(4 + 9i)(4 - 3i)$. To do this, we use the distributive property: $(4 + 9i)(4 - 3i) = 4(4) + 4(-3i) + 9i(4) + 9i(-3i) = 16 - 12i + 36i - Show moreā¦
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For some practice working with complex numbers: Calculate (2 + 2i) + (1 - 2i) = (2 + 2i) - (1 - 2i) = (2 + 2i)(1 - 2i) = The complex conjugate of (1 + i) is (1 - i). In general to obtain the complex conjugate reverse the sign of the imaginary part. (Geometrically this corresponds to finding the "mirror image" point in the complex plane by reflecting through the x-axis. The complex conjugate of a complex number z is written with a bar over it: Ź and read as "z bar". Notice that if z = a + ib, then (z)(Ź) = |z|Ā² = aĀ² + bĀ² which is also the square of the distance of the point z from the origin. (Plot z as a point in the "complex" plane in order to see this.) If z = 2 + 2i then Ź = and |z| = You can use this to simplify complex fractions. Multiply the numerator and denominator by the complex conjugate of the denominator to make the denominator real. 2 + 2i / 1 - 2i = Two convenient functions to know about pick out the real and imaginary parts of a complex number. Re(a + ib) = a (the real part (coordinate) of the complex number), and Im(a + ib) = b (the imaginary part (coordinate) of the complex number. Re and Im are linear functions -- now that you know about linear behavior you may start noticing it often.
Donna D.
Compute the real and imaginary parts of z = (i-4)/(2i-3). Compute the absolute value and the conjugate of z = (1+i)^6, w = i^17. Write in the "algebraic" form (a + ib) the following complex numbers z = i^5 + i + 1, w = (3+3i)^8. Write in the "trigonometric" form (rho(cos theta + i sin theta)) the following complex numbers a) 8 b) 6i c) (cos(pi/3) - i sin(pi/3))^7. 5. Simplify (a) (1+i)/(1-i) - (1+2i)(2+2i) + (3-i)/(1+i); (b) 2i(i-1) + (sqrt(3)+i)^3 + (1+i)(1+i)^-. 6. Compute the square roots of z = -1 - i. 7. Compute the cube roots of z = -8. 8. Prove that there is no complex number such that |z| - z = i.
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