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The easiest thing to do is to first express each number in the form $re^{i \\theta}$ where $r > 0$ and $\\theta \\in (-\\pi , \\pi]$. Our absolute value with then be equal to $r$, and the principal argument will be $\\theta$ (remember that the principal argument must takes values in $(-\\pi , \\pi]$).

\n

Consider, for example, the number $2e^{\\frac{3 \\pi}{2} i}$. We cannot take $\\theta = \\frac{3 \\pi}{2}$ because $\\frac{3 \\pi}{2}$ is not in $(-\\pi , \\pi]$. However, we know that $e^{\\frac{3 \\pi}{2} i} = e^{\\frac{3 \\pi}{2} i + 2k\\pi i}$ for all $k \\in \\mathbb{Z}$, and so $e^{\\frac{3 \\pi}{2} i} = e^{-\\frac{ \\pi}{2} i}$ (just take $k = -1$). Hence we take $\\theta = -\\frac{\\pi}{2}$. We can easily see that $r = 2$.

\n

So, we have already found the absolute value and the principal argument. In order to find the real part and the imaginary part, we make use of Euler's formula: For any real number $x$ we have $e^{x i} = \\cos(x) + i \\sin (x)$.

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Hence, we can see that $2e^{\\frac{3 \\pi}{2} i} = 2e^{-\\frac{ \\pi}{2} i} = 2\\cos(-\\frac{ \\pi}{2}) + 2i \\sin (-\\frac{ \\pi}{2}) = 0 -2i$. So, the real part is equal to $0$, and the imaginary part is equal to $-2$.

Sometimes, the $\\cos$ and the $\\sin$ will not evaluate to such nice numbers and you will have to keep your answers in terms of $\\cos$ and $\\sin$ to ensure that they are accurate answers. Also, your answers may need to include powers of $e$ in order to maintain accuracy.

\n

As another example, consider the number $-2e^{\\frac{\\pi}{2} i}$. We cannot take $r = -2$ because $-2$ is not greater than $0$. To get around this we use the following:
For any real number $x$ we have $-e^{x i} = (-1) \\cdot e^{x i} = e^{\\pi i} \\cdot e^{x i} = e^{(x+\\pi) i}$.
Hence, $-2e^{\\frac{\\pi}{2} i} = 2e^{\\frac{3 \\pi}{2} i}$. We can now proceed as we did in the first example.

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Find the real part, the imaginary part, the modulus, and the principal argument of each of the following complex numbers. If you require a hint, then press the \"Show steps\" button below.

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If you wish to write something like $\\sqrt{2}$ then please write 2^(1/2)
If you wish to write something like $e^{2}$ then please write e^(2).
If you wish to write something like $2 \\cos (3)$ then please write 2*cos(3).
If you wish to write $\\pi$ then please write pi, but make sure there are spaces on either side,
e.g. 2* pi *i.

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i) $\\var{{a}} e^{\\simplify{{b}/2} \\pi i}$
Real part: [[0]]
Imaginary part: [[1]]
Modulus: [[2]]
Argument: [[3]]

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ii) $-\\var{{c}} e^{\\simplify{{d}/3} \\pi i}$
Real part: [[4]]
Imaginary part: [[5]]
Modulus: [[6]]
Argument: [[7]]

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iii) $e^{\\simplify{{f} + {g}*i}}$
Real part: [[8]]
Imaginary part: [[9]]
Modulus: [[10]]
Argument: [[11]]

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iv) $e^{i (\\simplify{{h} + {k}*i})}$
Real part: [[12]]
Imaginary part: [[13]]
Modulus: [[14]]
Argument: [[15]]

Hint for part i: Suppose we have a complex number $x + iy$ where $x,y \\in \\mathbb{R}$. The modulus of this number is $\\sqrt{x^2 + y^2}$. To find the principal argument, we calculate $\\tan^{-1}(\\frac{y}{x})$, but remember that for this to be the principal argument it must be in $(-\\pi , \\pi]$. See the \"Show steps\" section of part a for an explanation on how to ensure that this is the case.

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Hint for part ii: In part i we found the modulii and principal arguments of $z_1$ and $z_2$, and we can express $z_1$ and $z_2$ as $z_1 = r_1 e^{\\theta_1}$ and $z_2 = r_2 e^{\\theta_2}$.

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We can see now that $z = \\frac{{z_1}^6}{{z_2}^{10}} = \\frac{{r_1}^6}{{r_2}^{10}} e^{(6 \\theta_1 - 10 \\theta_2 ) i}$. We can find the modulus easily from this. The same applies to the principal argument, although we must ensure it is in $(-\\pi , \\pi]$.

\n

In order to find the real and imaginary parts of $z$ we can apply our methods from part a (Look at the \"Show steps\" section of part a for more information).

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Let $z_1 : = \\var{latex(z1_re)} + \\var{latex(z1_im)} i$ and $z_2 : = \\var{latex(z2_re)} + \\var{latex(z2_im)} i$.

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i) What are the modulii and principal arguments of $z_1$ and $z_2$? If you require a hint, then press the \"Show steps\" button below.
$r_1 := \\| z_1 \\| =$ [[0]]
$\\theta_1 := \\arg (z_1) =$ [[1]]

$r_2 := \\| z_2 \\| =$ [[2]]
$\\theta_2 := \\arg (z_2) =$ [[3]]

\n

ii) Now calculate the real part, the imaginary part, the modulus, and the argument of $z := \\frac{{z_1}^6}{{z_2}^{10}}$. For the real and imaginary parts, you may need to express your answers using $\\sin$ and $\\cos$ in order to preserve accuracy. Press the \"Show steps\" button below if you require a hint.

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$\\textbf{Re} (z) =$ [[4]]
$\\textbf{Im} (z) =$ [[5]]
$\\| z \\| =$ [[6]]
$\\arg (z) =$ [[7]]

\n

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In this part of the question we will find the general real solution to the difference equation
$\\simplify{sub(y,x+2) + {coeff_1} * sub(y,x+1) + {coeff_2} * sub(y,x)} = \\var{{a1}} x + \\var{{a2}}$.

\n

\n

The homogeneous equation is
$\\simplify{sub(y,x+2) + {coeff_1} *sub(y,x+1) + {coeff_2} *sub(y,x)} = 0$.

\n

Hence, the auxiliary equation is (Please write it in terms of the letter $m$):
[[0]] $= 0$ .

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The roots of this are (please write the root with positive imaginary part first):
$m_1 =$ [[1]];
$m_2 =$ [[2]] .

\n

This implies that the complimentary sequence is
$(CS)_x = A \\Big($ [[1]] $\\Big)^x + \\bar{A} \\Big($ [[2]] $\\Big)^x$
Where $A$ is an arbitrary complex constant. Recall that the use of $A$ and $\\bar{A}$, instead of $A$ and $B$ (where $B$ is some other arbitary complex constant), ensures that our solution is real.

\n

Our complimentary sequence is implicitely real, but it is not explicitely real. That is, it will evaluate to a real value, but it still has imaginary numbers in its definition. We will now make it explicitely real. (You may wish to have a look at your answer to exercise 35.4.4 of the lectures notes, as this is a similar problem).

\n

What is the modulus and principal argument of our first root, $m_1$?
$\\| m_1 \\| =$ [[3]] ;
$\\arg (m_1) =$ [[4]] .

\n

Hence, the explicitely real complimentary sequence is
$(CS)_x = \\Big($ [[5]] $\\Big)^x \\cdot \\Big( C \\cos \\Big($ [[6]] $\\Big) + D \\sin \\Big($ [[6]] $\\Big) \\Big)$
where $C$ and $D$ are arbitrary real constants.

\n

\n

Now that we have the complimentary sequence, let us find the particular sequence. Given that the right hand side of the non-homogeneous equation is $\\var{{a1}} x + \\var{{a2}}$, we should try $y_x = ax +b$, where $a$ and $b$ are constants to be found. Note that we then have $y_{x+1} = a(x+1) +b$, $y_{x+2} = a(x+2) +b$, and so on.

\n

By substituting $y_x = ax +b$ into our non-homogeneous equation, $\\simplify{sub(y,x+2) + {coeff_1} *sub(y,x+1) + {coeff_2} *sub(y,x)} = \\var{{a1}} x + \\var{{a2}}$, and comparing 1) the coefficients of $x$ on both sides, and 2) the constant coefficients on both sides, we can find $a$ and $b$. (Please write your answers as single fractions; e.g. if you wish to write $\\frac{3}{2} + \\frac{5}{3}$ then write $\\frac{19}{6}$, otherwise the system will not correctly interpret what you wish to write).
$a =$ [[7]]
$b =$ [[8]]

\n

Hence, our particular solution is
$(PS)_x = \\Big($ [[7]] $\\Big) \\cdot x + \\Big($ [[8]] $\\Big)$.

\n

Therefore, the general solution to the non-homogeneous equation is
$(CS)_x + (PS)_x = \\Big($ [[5]] $\\Big)^x \\cdot \\Big( C \\cos \\Big($ [[6]] $\\Big) + D \\sin \\Big($ [[6]] $\\Big) \\Big) + \\Big($ [[7]] $\\Big) \\cdot x + \\Big($ [[8]] $\\Big)$.

\n

This is the second root of our auxilliary equation. It is dependent (via the variable random) on the argument of the roots - the variable theta - and the variable r.

", "group": "Variables for part c"}, "theta_1": {"definition": "switch(random_1 = 1 , pi/6 , random_1 = 2 , pi/3 , random_1 = 3 , pi/4, 0)", "templateType": "anything", "name": "theta_1", "description": "

This is the argument of the number z1 in part b. It is dependent on the random variable random_1.

", "group": "Variables for part b"}, "z_abs": {"definition": "abs(z)", "templateType": "anything", "name": "z_abs", "description": "

This is the absolute value of z.

", "group": "Variables for part b"}, "r1": {"definition": "random(1..6 except(2))", "templateType": "anything", "name": "r1", "description": "

This is the modulus of the number z1 in part b.

", "group": "Variables for part b"}, "a2": {"definition": "random(2..6)", "templateType": "anything", "name": "a2", "description": "

This is the constant coefficient in the right-hand-side of our non-homogeneous difference equation.

", "group": "Variables for part c"}, "a1": {"definition": "random(2..6)", "templateType": "anything", "name": "a1", "description": "

This is the coefficient of x in the right-hand-side of our non-homogeneous difference equation.

", "group": "Variables for part c"}, "random_1": {"definition": "random(1..3)", "templateType": "anything", "name": "random_1", "description": "

This is a randomly generated number which helps to determine the argument, theta_1, of the number z1 in part b.

", "group": "Variables for part b"}, "n1_abs": {"definition": "abs(n1)", "templateType": "anything", "name": "n1_abs", "description": "

This is the absolute value of n1.

", "group": "Variables for part a"}, "random_2": {"definition": "random(1..3)", "templateType": "anything", "name": "random_2", "description": "

This is a randomly generated number which helps to determine the argument, theta_2, of the number z2 in part b.

", "group": "Variables for part b"}, "a": {"definition": "random(2..6)", "templateType": "anything", "name": "a", "description": "

This will be used in the definition of one of our four variables, n1, n2, n3, n4.

", "group": "Variables for part a"}, "z2_re": {"definition": "switch(random_2 = 1 , latex(\"\\\\simplify\\{\\{r2\\}/2\\} \\\\sqrt\\{3\\}\") ,random_2 = 2 , latex(\"\\\\simplify\\{\\{r2\\}/2\\}\") , random_2 = 3 , latex(\"\\\\simplify\\{\\{r2\\}/2\\} \\\\sqrt\\{2\\}\") , 0 )", "templateType": "anything", "name": "z2_re", "description": "

This is the real part of the number z2 in part b. It has been expressed in latex code as there is no other way (that I know) to express the square-root symbol. Otherwise the system would approximate it as a fraction.

", "group": "Variables for part b"}, "n4": {"definition": "(e^(-k + h * i))", "templateType": "anything", "name": "n4", "description": "

This is our fourth number in part a.

", "group": "Variables for part a"}, "z1_im": {"definition": "switch(random_1 = 1 , latex(\"\\\\simplify\\{\\{r1\\}/2\\}\") ,random_1 = 2 , latex(\"\\\\simplify\\{\\{r1\\}/2\\} \\\\sqrt\\{3\\}\") , random_1 = 3 , latex(\"\\\\simplify\\{\\{r1\\}/2\\} \\\\sqrt\\{2\\}\") , 0 )", "templateType": "anything", "name": "z1_im", "description": "

This is the imaginary part of the number z1 in part b. It has been expressed in latex code as there is no other way (that I know) to express the square-root symbol. Otherwise the system would approximate it as a fraction.

", "group": "Variables for part b"}, "n3_re": {"definition": "re(n3)", "templateType": "anything", "name": "n3_re", "description": "

This is the real part of n3.

", "group": "Variables for part a"}, "f": {"definition": "random(1..5)", "templateType": "anything", "name": "f", "description": "

This will be used in the definition of one of our four variables, n1, n2, n3, n4.

", "group": "Variables for part a"}, "coeff_1": {"definition": "switch(random = 1 , -r , random = 2 , r , 0)", "templateType": "anything", "name": "coeff_1", "description": "

This is the first coefficient of our auxilliary equation. It is dependent (via the variable random) on the argument of the roots that we have defined - the variable theta - and the variable r.

", "group": "Variables for part c"}, "coeff_2": {"definition": "switch(random = 1 , r^2 , random = 2 , r^2 , 0)", "templateType": "anything", "name": "coeff_2", "description": "

This is the second coefficient of our auxilliary equation. It is dependent (via the variable random) on the argument of the roots that we have defined - the variable theta - and the variable r.

", "group": "Variables for part c"}, "n3_im": {"definition": "im(n3)", "templateType": "anything", "name": "n3_im", "description": "

This is the imaginary part of n3.

", "group": "Variables for part a"}, "n3_abs": {"definition": "abs(n3)", "templateType": "anything", "name": "n3_abs", "description": "

This is the absolute value of n3.

", "group": "Variables for part a"}, "n3": {"definition": "(e^(f + g * i))", "templateType": "anything", "name": "n3", "description": "

This is our third number in part a.

", "group": "Variables for part a"}, "n4_arg": {"definition": "arg(n4)", "templateType": "anything", "name": "n4_arg", "description": "

This is the argument of n4.

", "group": "Variables for part a"}, "PI_2": {"definition": "switch(random = 1 , a2/(r^2 - r + 1) - a1 * (2 - r)/((r^2 - r + 1)^2) , random = 2 , a2/(r^2 + r + 1) - a1 * (2 + r)/((r^2 + r + 1)^2) , 0)", "templateType": "anything", "name": "PI_2", "description": "

This is the constant coefficient in our particular solution.

", "group": "Variables for part c"}, "d": {"definition": "random(1..6)", "templateType": "anything", "name": "d", "description": "

This will be used in the definition of one of our four variables, n1, n2, n3, n4.

", "group": "Variables for part a"}, "n2_arg": {"definition": "arg(n2)", "templateType": "anything", "name": "n2_arg", "description": "

This is the argument of n2.

", "group": "Variables for part a"}, "n4_re": {"definition": "re(n4)", "templateType": "anything", "name": "n4_re", "description": "

This is the real part of n4.

", "group": "Variables for part a"}, "n2_re": {"definition": "re(n2)", "templateType": "anything", "name": "n2_re", "description": "

This is the real part of n2.

", "group": "Variables for part a"}, "PI_1": {"definition": "switch(random = 1 , a1/(r^2 - r + 1) , random = 2 , a1/(r^2 + r + 1) , 0)", "templateType": "anything", "name": "PI_1", "description": "

This is the coefficient of x in our particular solution.

", "group": "Variables for part c"}, "n4_im": {"definition": "im(n4)", "templateType": "anything", "name": "n4_im", "description": "

This is the imaginary part of n4.

", "group": "Variables for part a"}, "theta_2": {"definition": "switch(random_2 = 1 , pi/6 , random_2 = 2 , pi/3 , random_2 = 3 , pi/4, 0)", "templateType": "anything", "name": "theta_2", "description": "

This is the argumentof the number z2 in part b. It is dependent on the random variable random_2.

", "group": "Variables for part b"}, "n1_re": {"definition": "re(n1)", "templateType": "anything", "name": "n1_re", "description": "

This is the real part of n1.

", "group": "Variables for part a"}, "c": {"definition": "random(2..6)", "templateType": "anything", "name": "c", "description": "

This will be used in the definition of one of our four variables, n1, n2, n3, n4.

", "group": "Variables for part a"}, "b": {"definition": "random(1..4 except(2))", "templateType": "anything", "name": "b", "description": "

This will be used in the definition of one of our four variables, n1, n2, n3, n4.

", "group": "Variables for part a"}, "z_re": {"definition": "re(z)", "templateType": "anything", "name": "z_re", "description": "

This is the real part of z.

", "group": "Variables for part b"}, "n3_arg": {"definition": "arg(n3)", "templateType": "anything", "name": "n3_arg", "description": "

This is the argument of n3.

", "group": "Variables for part a"}, "n2": {"definition": "-c * (e^(Pi * d * i / 3))", "templateType": "anything", "name": "n2", "description": "

This is our second number in part a.

", "group": "Variables for part a"}, "z_im": {"definition": "im(z)", "templateType": "anything", "name": "z_im", "description": "

This is the imaginary part of z.

", "group": "Variables for part b"}, "z_arg": {"definition": "arg(z)", "templateType": "anything", "name": "z_arg", "description": "

This is the argument of z.

", "group": "Variables for part b"}, "n1": {"definition": "a * (e^(Pi * b * i / 2))", "templateType": "anything", "name": "n1", "description": "

This is our first number in part a.

", "group": "Variables for part a"}, "root_1": {"definition": "r * switch(random = 1 , 1/2 + (sqrt(3))*i/2 , random = 2 , -1/2 + (sqrt(3))*i/2 , 0)", "templateType": "anything", "name": "root_1", "description": "

This is the first root of our auxilliary equation. It is dependent (via the variable random) on the argument of the roots - the variable theta - and the variable r.

", "group": "Variables for part c"}, "z1_re": {"definition": "switch(random_1 = 1 , latex(\"\\\\simplify\\{\\{r1\\}/2\\} \\\\sqrt\\{3\\}\") ,random_1 = 2 , latex(\"\\\\simplify\\{\\{r1\\}/2\\}\") , random_1 = 3 , latex(\"\\\\simplify\\{\\{r1\\}/2\\} \\\\sqrt\\{2\\}\") , 0 )", "templateType": "anything", "name": "z1_re", "description": "

This is the real part of the number z1 in part b. It has been expressed in latex code as there is no other way (that I know) to express the square-root symbol. Otherwise the system would approximate it as a fraction.

", "group": "Variables for part b"}, "theta": {"definition": "switch(random = 1 , pi/3 , random = 2 , 2*pi/3 , 0)", "templateType": "anything", "name": "theta", "description": "

This is the argument of the first root of our auxilliary equation. It is dependent on the variable random.

", "group": "Variables for part c"}, "r2": {"definition": "random(1..6 except(2))", "templateType": "anything", "name": "r2", "description": "

This is the modulus of the number z2 in part b.

", "group": "Variables for part b"}, "random": {"definition": "random(1, 2)", "templateType": "anything", "name": "random", "description": "

This is a random variable which determines the argument of the roots of our auxilliary equation.

", "group": "Variables for part c"}, "z": {"definition": "(((r1)^6)/((r2)^10)) * e^((6 * theta_1 - 10 * theta_2) * i)", "templateType": "anything", "name": "z", "description": "

This is z1 ^ 6 / z2 ^ 10.

", "group": "Variables for part b"}, "g": {"definition": "random(0..5)", "templateType": "anything", "name": "g", "description": "

This will be used in the definition of one of our four variables, n1, n2, n3, n4.

", "group": "Variables for part a"}, "k": {"definition": "random(1..5)", "templateType": "anything", "name": "k", "description": "

This will be used in the definition of one of our four variables, n1, n2, n3, n4.

", "group": "Variables for part a"}, "r": {"definition": "random(4 , 6 , 8 , 10 , 12)", "templateType": "anything", "name": "r", "description": "

This is the modulus of the roots of our auxilliary equation.

", "group": "Variables for part c"}, "n1_im": {"definition": "im(n1)", "templateType": "anything", "name": "n1_im", "description": "

This is the imaginary part of n1.

", "group": "Variables for part a"}, "z2_im": {"definition": "switch(random_2 = 1 , latex(\"\\\\simplify\\{\\{r2\\}/2\\}\") ,random_2 = 2 , latex(\"\\\\simplify\\{\\{r2\\}/2\\} \\\\sqrt\\{3\\}\") , random_2 = 3 , latex(\"\\\\simplify\\{\\{r2\\}/2\\} \\\\sqrt\\{2\\}\") , 0 )", "templateType": "anything", "name": "z2_im", "description": "

This is the imaginary part of the number z2 in part b. It has been expressed in latex code as there is no other way (that I know) to express the square-root symbol. Otherwise the system would approximate it as a fraction.

", "group": "Variables for part b"}, "n2_im": {"definition": "im(n2)", "templateType": "anything", "name": "n2_im", "description": "

This is the imaginary part of n2.

", "group": "Variables for part a"}, "n2_abs": {"definition": "abs(n2)", "templateType": "anything", "name": "n2_abs", "description": "

This is the absolute value of n2.

", "group": "Variables for part a"}, "h": {"definition": "random(1..5)", "templateType": "anything", "name": "h", "description": "

This will be used in the definition of one of our four variables, n1, n2, n3, n4.

", "group": "Variables for part a"}, "n1_arg": {"definition": "arg(n1)", "templateType": "anything", "name": "n1_arg", "description": "

This is the argument of n1.

", "group": "Variables for part a"}, "n4_abs": {"definition": "abs(n4)", "templateType": "anything", "name": "n4_abs", "description": "

This is the absolute value of n4.

", "group": "Variables for part a"}}, "advice": "", "variablesTest": {"condition": "", "maxRuns": 100}, "name": "MA100 LT Week 8", "statement": "

Lent Term Week 8 (lectures 35 and 36): In this question you will look at complex numbers, and second order recurrence relations where the auxiliary equation has complex roots.

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", "preamble": {"js": "", "css": ""}, "tags": [], "rulesets": {}, "ungrouped_variables": [], "metadata": {"description": "

This is the question for Lent Term week 8 of the MA100 course at the LSE. It looks at material from chapters 35 and 36.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "extensions": [], "type": "question", "contributors": [{"name": "Michael Yiasemides", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1440/"}]}]}], "contributors": [{"name": "Michael Yiasemides", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1440/"}]}