Recent questions tagged complex

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What is the significance of complex integration along a contour?
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Define a complex number and its basic properties.
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What is the conjugate of the complex number \(7+3i\) ?
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Suppose $f, g$ are entire functions, and suppose that for all $z \in \mathbb{C}$, that $| f(z) | \leq | g(z)|$. What conclusion can you draw?
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explain why $z1 \times z2 = (\bar{z1}\times z2 + z1 \times \bar{z2})/2
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Simplify the complex expression\[(2-3 i)(-2+i)\]
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Simplify the complex number \(5(3-2 i)+2 i(4+6 i)\)
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Which expression is equivalent to \(9 x^2-16 y^2 ?\)
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(Lael and Nourouzi 2008) Let \((V, \mu, \nu)\) be an \(I F\)-normed space. Assume further that \(\mu(x, t)>0\) for all \(t>0\) implies \(x=0\). Define\[\|x\|_\alpha=\inf ...
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Give an example of how to find the roots of a complex number. use latex
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What is a complex root? use latex
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What are the basic operations of complex numbers? Use latex and give examples for each.
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Explain the polar form of complex numbers with an example and use latex
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What happens when you multiply a complex number by its conjugate? Give an example and use latex
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Give 10 examples of complex conjugate pairs. use latex
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What is conjugation in complex numbers? use latex
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WHat are complex numbers? use latex
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The inverse of \(-i\) in the multiplicative group, \(1,-1, i,-i\) is
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What is a zero-matrix?
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Multiply \(3+\mathrm{i} \sqrt{3}\) by its complex conjugate.
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For \(\mathrm{i}=\sqrt{-1}\), what is the result of subtracting \((2+4 \mathrm{i})\) from \((-5+6 \mathrm{i}) ?\)
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What is a CCP Complex?
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What is a complex number?
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What is the formula for the area of a parallelogram?
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Prove that the roots of a polynomial with real coefficients are either real or come in conjugate pairs.
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If \(x\) is a real number such that \((x-3)(x-1)(x+1)(x+3)+16=116^2\), what is the largest possible value of \(x\) ?
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Mr. Jones teaches algebra. He has a whiteboard with a pre-drawn coordinate grid that runs from \(-10\) to 10 in both the \(x\) and \(y\) coordinates. Consequently, when h...
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Say that a sequence \(a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8\) is cool if- the sequence contains each of the integers 1 through 8 exactly once, and- every pair of consecu...
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Let \(H\) be a regular hexagon with area 360 . Three distinct vertices \(X, Y\), and \(Z\) are picked randomly, with all possible triples of distinct vertices equally lik...
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The roots of the polynomial \(x^4-4 i x^3+3 x^2-14 i x-44\) form the vertices of a parallelogram in the complex plane. What is the area of the parallelogram?
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The imaginary part of \((3+2 \sqrt{-54})^{\frac{1}{2}}-(3-2 \sqrt{-54})^{\frac{1}{2}}\) can be(1) \(\sqrt{-6}\)(2) \(\sqrt{6}\)(3) \(-2 \sqrt{6}\)(4) 6
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We have a chessboard of size \(\left(k^2-k+1\right) \times\left(k^2-k+1\right)\), where \(k-1=p\) is a prime number. For each prime \(p\), give a method of distribution o...
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The positive integers \(a, b, c, d, p\), and \(q\) satisfy \(a d-b c=1\) and \(\frac{a}{b}>\frac{p}{q}>\frac{c}{d}\). Prove that: (a) \(q \geq b+d\);(b) If \(q=b+d\), the...
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Let \(x \leq y \leq z\) be real numbers such that \(x y+y z+z x=1\). Prove that \(x z<\frac{1}{2}\). Is it possible to improve the value of constant \(\frac{1}{2}\) ?
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Let \(z_1, z_2, z_3\) be pairwise distinct complex numbers satisfying \(\left|z_1\right|=\left|z_2\right|=\left|z_3\right|=\) 1 and\[\frac{1}{2+\left|z_1+z_2\right|}+\fra...
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If \(a, b, c\) are nonnegative real numbers with \(a^2+b^2+c^2=1\), prove that\[\frac{a}{b^2+1}+\frac{b}{c^2+1}+\frac{c}{a^2+1} \geq \frac{3}{4}(a \sqrt{a}+b \sqrt{b}+c \...
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For any set \(A\) of positive integers, let \(n_A\) be the number of triples \((x, y, z)\) of elements of \(A\) with \(x<y\) and \(x+y=z\). If \(A\) is a seven-element se...
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Messages are coded as sequences of zeros and ones. Only sequences with not more than two consecutive zeros or ones are allowed. How many permitted 12digit sequences are t...
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Let \(T(n)\) be the sum of the decimal digits of a natural number \(n\). (a) Find a natural number \(N\) such that \(T(k N)\) is even for all \(k=1,2, \ldots, 1992\), but...
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Let \(s(t)\) denote the number of digits of a natural number \(t\). Find all solutions to the system\[\begin{aligned}s(x)+s(y) &=x \\x+y+s(z) &=z \\s(x)+s(y)+s(z) &=y-4\e...
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Let \(a, b\) and \(c\) be positive real numbers. Prove that\[\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \leq \frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2} .\]
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If \(a, b, c\) are positive real numbers that satisfy\[\frac{a^2}{1+a^2}+\frac{b^2}{1+b^2}+\frac{c^2}{1+c^2}=1\]prove that \(|a b c| \leq \frac{1}{2 \sqrt{2}}\)
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If \(a, b, c\) are positive real numbers that satisfy \(a^2+b^2+c^2=1\), find the minimal value of\[S=\frac{a^2 b^2}{c^2}+\frac{b^2 c^2}{a^2}+\frac{c^2 a^2}{b^2}\]
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Given that \(z=3-3 \mathrm{i}\) express, in the form \(a+\mathrm{i} b\), where \(a\) and \(b\) are real numbers,(a) \(z^2\),(b) \(\frac{1}{z}\).(c) Find the exact value o...
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The complex numbers \(z_1\) and \(z_2\) are given by\[\begin{aligned}&z_1=5+3 \mathrm{i} \\&z_2=1+p \mathrm{i}\end{aligned}\]where \(p\) is an integer.(a) Find \(\frac{\m...
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(a) Find the roots of the equation\[z^2+2 z+17=0,\]giving your answers in the form \(a+\mathrm{i} b\), where \(a\) and \(b\) are integers.(b) Show these roots on an Argan...
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\[z=\sqrt{3}-\mathrm{i} .\]\(\mathrm{z}^*\) is the complex conjugate of \(\mathrm{z}\).(a) Show that \(\frac{z}{z^*}=\frac{1}{2}-\frac{\sqrt{3}}{2} i\).(b) Find the value...
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The complex number \(z\) is defined by\[z=\frac{a+2 \mathrm{i}}{a-\mathrm{i}}, a \in \Re, a>0\]Given that the real part of \(z\) is \(\frac{1}{2}\), find(a) the value of ...
287
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The complex numbers \(z_1\) and \(z_2\) are given by\[z_1=2-\mathrm{i} \text { and } z_2=-8+9 \mathrm{i}\](a) Show \(z_1\) and \(z_2\) on a single Argand diagram.Find, sh...
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The complex numbers \(z_1\) and \(z_2\) are given by\[z_1=2+8 \mathrm{i} \text { and } z_2=1-\mathrm{i}\]Find, showing your working,(a) \(\frac{z_1}{z_2}\) in the form \(...
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