28 Punchline Algebra Book A Answer Key
Dionisio Parmar
Self learning Linear Algebra I am having some difficulty in understanding left and right inverses of a given algebraic operation. So the operation is $x\circ y=x^y$ for all positive numbers.In order to find the inverses I have split the operation into 2 separate equations of form$a\circ u=b$ and $v\circ a=b$ and solving $u$ and $v$ for some $a$ and $b$ from the set of positive numbers. This approach gave me right inverse as $\log_xy$ and left inverse as $x\sqrty$. Is this solution correct?
The punchline is that you can indeed take out the factor of two from the brackets before going to polar form under the caveat that (as geetha290krm noted) the power is an integer. If it is not, you will have to take more care.
Homework will be due once per week. This page will be updated every time a new problem set is posted. Solutions for each problem set will be posted on this page after the assignment is due. assignments Assignment 1 (due Friday, September 16 at 4pm)Instructions: Your answers should always be written in narrative form; you should of course include relevant computations and equations, but these should be situated in the framework of some overall narrative that explains your reasoning, and they should always be naturally integrated into the structure of your writing. You should follow standard rules of composition: write in complete sentences, include appropriate punctuation, group sentences with a common theme into a paragraph, provide appropriate narrative transition when your reasoning takes a significant turn, etc. When writing formal proofs, you should not use abbreviations like $\forall$, $\exists$, $\Rightarrow$. It is acceptable to use the notation $\in$ when describing elements in a formal proof (e.g., "Since $2 \in \mathbbZ$, we see that ..."). As a general rule, a mathematical symbol should never start a sentence or follow punctuation; the exception is that you can write a mathematical symbol after a colon which is announcing the beginning of a list of mathematical objects. Of course your work should be legible and neat. The overall guiding principle in your writing is to remember that your work is meant to be read by a a skeptical peer; your job is to write a (logically) convincing argument to this audience.
Though complex numbers now enjoyed a well-earned reputation for being applicable in the "real world," for a long time they were viewed with a lot of suspicion and mistrust. (For that matter, negative numbers were i>verbotten considered to be just as ridiculous as well!). The ice began to thaw on this exclusion when Italian mathematicians began to make considerable progress on solving the general cubic equation in the 1500s. For instance, using the work of Scipione del Ferro and Niccolo Tartaglia, Gerolamo Cardano was able to provide an equation for solving cubic equations of the form $x^3=ax+b$, where $a,b >0$: $$x = \sqrt[3]\sqrt-\left(\fraca3\right)^3+\left(\fracb2\right)^2+\fracb2 - \sqrt[3]\sqrt-\left(\fraca3\right)^3+\left(\fracb2\right)^2-\fracb2.$$ The issue with this formula is that for certain choices of $a$ and $b$, the quantity inside the square root becomes negative, and so one is left dealing with (what we now call) a complex number. Curiously, then, the real solutions to such equations seem to need complex numbers in order to be computed. This was one of the driving forces that helped convince mathematicians that complex numbers were deeply meaningful tools for answering classical questions.
Prove that $\mathbbC$ is not ordered. That is, show that there is no set $\mathbbP \subseteq \mathbbC - \0\$ which is closed under addition and multiplication , and such that for any $z \in \mathbbC$, precisely one of the following statements holds: $z \in \mathbbP$, or $z=0$, or $-z \in \mathbbP$. [Note: make sure you are extremely careful in your proof. You may use the fact that $\mathbbC$ is a field without proof, but you must explicitly cite any usage of field axioms as you use them.]
- Find $a,b \in \mathbbR$ so that $\displaystyle \frac3+2ii-4+\fraci^3+2-i-1=a+bi$.
- Let $a,b,c \in \mathbbR$ be given, and suppose that $w = x+yi$ is a solution to the cubic equation $z^3+az^2+bz+c=0$. Prove that $\overlinew = x-yi$ is also a solution to this equation.
- Find three distinct complex solutions to the equation $z^3-3z^2+3z-2$. [Hint: Assume that $w = x+yi$ is a solution, and analyze what the vanishing $w^3-3w^2+3w-2$ means in terms of real and imaginary parts. You might also find the rational root theorem useful in making educated guesses about roots of certain real cubic equations that emerge from your analysis.]
[NB: You will show that this set is nonempty for any choice of $w \in \mathbbC-\0\$; this means that $\textexp$ surjects onto $\mathbbC-\0\$.]
- In this problem, we'll be exploring the connection between the exponential function and the basic trigonometric functions.
- Let $x \in \mathbbR$ be given. Express $\sin(x)$ and $\cos(x)$ as linear combinations of $e^ix$ and $e^-ix$. [Hint: Euler]
- Use your expressions to prove the familiar results that $\sin(x) \leq 1$ and $\cos(x)\leq 1$. [Hint: triangle inequality]
- Replace "$x$" with "$z$" in your answer to part (a). You now have functions which act as generalizations of the usual real trigonometric functions you are used to. We will abuse notation slightly and keep their same names; for example, we will still call the former $\sin$. Even though these functions are defined in a way that generalizes the usual definitions, we'll see that some familiar properties of these trig functions no longer hold (while others do). For instance, prove that $\sin:\mathbbC \to \mathbbC$ and $\cos:\mathbbC \to \mathbbC$ are unbounded.
- Prove that for all $z \in \mathbbC$ we have $\left(\sin(z)\right)^2+\left(\cos(z)\right)^2=1$.
- Prove that for any $z \in \mathbbC$ we have $\sin(-z)=-\sin(z)$ and $\cos(-z)=\cos(z)$.
- Prove that for any $z_1,z_2 \in \mathbbC$ we have $\sin(z_1+z_2)=\sin(z_1)\cos(z_2)+\cos(z_1)\sin(z_2)$. [NB: Problem (c) shows us that one of the familiar properties of the real functions $\sin$ and $\cos$ is no longer valid when we consider their complex generalizations. On the other hand, problems (d)--(f) show us that several other properties --- namely the Pythagorean formula, the oddness and evenness of the functions, and angle sum formulas --- do carry over into the complex setting.]
Here are the Solutions. Assignment 4 (due Friday, October 7 at 4pm)Instructions: Your answers should always be written in narrative form; you should of course include relevant computations and equations, but these should be situated in the framework of some overall narrative that explains your reasoning, and they should always be naturally integrated into the structure of your writing. You should follow standard rules of composition: write in complete sentences, include appropriate punctuation, group sentences with a common theme into a paragraph, provide appropriate narrative transition when your reasoning takes a significant turn, etc. When writing formal proofs, you should not use abbreviations like $\forall$, $\exists$, $\Rightarrow$. It is acceptable to use the notation $\in$ when describing elements in a formal proof (e.g., "Since $2 \in \mathbbZ$, we see that ..."). As a general rule, a mathematical symbol should never start a sentence or follow punctuation; the exception is that you can write a mathematical symbol after a colon which is announcing the beginning of a list of mathematical objects. Of course your work should be legible and neat. The overall guiding principle in your writing is to remember that your work is meant to be read by a a skeptical peer; your job is to write a (logically) convincing argument to this audience.
dd2b598166