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Complex Numbers

/Tag: Complex Numbers
11 02, 2017

Complex Proofs

Complex Proofs $z\in C$ such that $\frac{z}{z-i}$ is real. Show that z is imaginary. Let $\frac{z}{z-i}=a$, where a is real. Then $z=a(z-i)$ $z=az-ai$ $ai=az-z$ $ai=z(a-1)$ so $z=\frac{ai}{a-1}$ $z=\frac{a}{a-1}i$ Since a is real, then $\frac{a}{a-1}$ is real, and hence $\frac{a}{a-1}i$ is imaginary thus z is imaginary     [...]

6 02, 2017

Argand Diagrams

Argand Diagrams If $z=a+b i$, then its conjugate $\overline{z}=a-b i$ the conjugate of z is reflected in the Real axis worked examples further examples = 10x + 15 - 7x + 14 = 3x + 29 harder examples u = 1 + [...]

3 01, 2017

Complex Conjugate Roots Theorem

Complex Conjugate Roots Theorem If ax2 + bx +c = 0 has real coefficients, and a + bi is a root, with a and b real numbers, then its complex conjugate a − bi is also a root. Complex Conjugate Roots Theorem Proof Prove that if $\alpha$ is a complex  root  of ax2 [...]

2 01, 2017

Dividing Complex Numbers

Dividing Complex Numbers $\overline{a+ib}=a-ib$ worked examples Simplify (7 + 2i) ÷(3 + 3i). $=\frac{{7 + 2i}}{{3 + 3i}}$ rewrite in fraction form to make it simpler to see and work with $=\frac{{7 + 2i}}{{3 + 3i}} \times \frac{{3 - 3i}}{{3 - 3i}}$ multiply by the conjugate of the denominator [...]
Complex Numbers $z=a + bi$ Go to the Complex Numbers forum to see more questions or ask your own Topics Covered in Complex Numbers Operations on Complex Numbers Division Of Complex Numbers Conjugate Proofs Complex Conjugate Roots Theorem Argand Diagrams Operations [...]