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 $(-\\pi , \\pi]$).
\nConsider, 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 $r = 2$.
\nSo, 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 $e^{x i} = \\cos(x) + i \\sin (x)$.
\nHence, 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.
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.
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.
\nIf 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.
i) $\\var{{a}} e^{\\simplify{{b}/2} \\pi i}$
Real part: [[0]]
Imaginary part: [[1]]
Modulus: [[2]]
Argument: [[3]]
ii) $-\\var{{c}} e^{\\simplify{{d}/3} \\pi i}$
Real part: [[4]]
Imaginary part: [[5]]
Modulus: [[6]]
Argument: [[7]]
iii) $e^{\\simplify{{f} + {g}*i}}$
Real part: [[8]]
Imaginary part: [[9]]
Modulus: [[10]]
Argument: [[11]]
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.
\nHint 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 $z_2 = r_2 e^{\\theta_2}$.
\nWe 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 $(-\\pi , \\pi]$.
\nIn 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).
", "type": "information", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "marks": 0, "scripts": {}, "extendBaseMarkingAlgorithm": true, "unitTests": [], "customMarkingAlgorithm": ""}], "prompt": "Let $z_1 : = \\var{latex(z1_re)} + \\var{latex(z1_im)} i$ and $z_2 : = \\var{latex(z2_re)} + \\var{latex(z2_im)} i$.
\ni) 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]]
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.
\n$\\textbf{Re} (z) =$ [[4]]
$\\textbf{Im} (z) =$ [[5]]
$\\| z \\| =$ [[6]]
$\\arg (z) =$ [[7]]
In this part of the question we will find the general real solution to the difference equation
$\\simplify{sub(y,x+2) + {coeff_1} * sub(y,x+1) + {coeff_2} * sub(y,x)} = \\var{{a1}} x + \\var{{a2}}$.
The homogeneous equation is
$\\simplify{sub(y,x+2) + {coeff_1} *sub(y,x+1) + {coeff_2} *sub(y,x)} = 0$.
Hence, the auxiliary equation is (Please write it in terms of the letter $m$):
[[0]] $= 0$ .
The roots of this are (please write the root with positive imaginary part first):
$m_1 =$ [[1]];
$m_2 =$ [[2]] .
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.
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).
\nWhat is the modulus and principal argument of our first root, $m_1$?
$\\| m_1 \\| =$ [[3]] ;
$\\arg (m_1) =$ [[4]] .
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.
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.
\nBy 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]]
Hence, our particular solution is
$(PS)_x = \\Big($ [[7]] $\\Big) \\cdot x + \\Big($ [[8]] $\\Big)$.
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]] $\\Big)$.
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.
\nIf you have not provided an answer to every input gap of a question or part of the question, and you try to submit your answers to the question or part, then you will see the message \"Can not submit answer - check for errors\". In reality your answer has been submitted, but the system is just concerned that you have not submitted an answer to every input gap. For this reason, please ensure that you provide an answer to every input gap in the question or part before submitting. Even if you are unsure of the answer, write down what you think is most likely to be correct; you can always change your answer or retry the question.
\nAs with all questions, there may be parts where you can choose to \"Show steps\". This may give a hint, or it may present sub-parts which will help you to solve that part of the question. Furthermore, remember to always press the \"Show feedback\" button at the end of each part. Sometimes, helpful feedback will be given here, and often it will depend on how correctly you have answered and will link to other parts of the question. Hence, always retry the parts until you obtain full marks, and then look at the feedback again.
Keep in mind that in order to see the feedback for a particular part of a question, you must provide a full (but not necessarily correct) answer to that part. Do not worry though, as you can look at the feedback and then ammend your answer accordingly.
Furthermore, as with all questions, choosing to reveal the answers will only show you the answers which change every time the question is loaded (i.e. answers to randomised questions); the fixed answers will not be revealed.
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.
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