introduction about numbers

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Cool Stuff (Advanced) Exercise 21 (a bit tricky) Write $\cos(3\theta)$ in terms of $\cos(\theta)$. You can use the same method to express $\cos(n\theta)$ or $\sin(n\theta)$ as a polynomial in $\cos\theta$ or $\sin\theta$. Exercise 13 What is $|z|-|\textrm{Conj}(z)|$? + \dots$$ What happens if we use this power series to calculate $e^{i\theta}$? Similarly, taking the argument of both sides, we get $\arg(z) = \arg(z /w )+\arg(w)$ and so $\arg(z /w ) = \arg(z)- \arg(w)$. Although sin appears to be allowed to interfere with what God is doing, it does not ultimately triumph. I glossed over this problem earlier, but the argument of a complex number is not unique, the argument of $i$ could be $90^{\circ}$ or it could be $450^{\circ}$ , or in fact $(90+360n)^{\circ}$ for any integer $n$. In other words, we take one complex number to the power of another complex number and we get a real number! Some functions have what is called a power series . Introduction: What is a Number System? Going back to our old friend the quadratic equation, if $a$, $b$ and $c$ are real numbers, the solution to $a x^2 + b x + c = 0$ is: $$ x = \frac { -b\pm \sqrt{b^2 - 4a c}} {2a} $$ But now this formula works if $b^2 - 4ac < 0$ . The Book of Numbers (from Greek Ἀριθμοί, Arithmoi; Hebrew: בְּמִדְבַּר ‎, Bəmiḏbar, "In the desert [of]") is the fourth book of the Hebrew Bible, and the fourth of five books of the Jewish Torah. We can't multiply the top and bottom by $i$ now, because the bottom would still be a complex number which is no good. If one understands the numbers to be literal and the men to represent about one-fourth of the population, then the number of the Israelites ranges from two to three million people10A literal understanding of the numbers in the census is in congruence with Pharaoh’s fear of the rapidly increasing Hebrews overrunning Egypt (Ex 1:7-12), the promises made to Abraham about becoming a great nation (Gen 12:2; 17:5-6), the earlier census taken during the first year in the wilderness (Exod 30:12--16; 38:26), and other traditions about the numbers of adult males who left Egypt (Ex 12:37; Num 11:21)11. b. Multiplying this by $\alpha$, we get $\alpha^3 = -5\alpha$ . Their position as covenantal people obligates them to subject the whole area of their life under the control of God: worship, social, family, and individual. 21 Although some of the didactic sections are parallel with Leviticus (e.g., the prescription concerning the seasonal feasts in 28; 29; cf. - \frac{\theta^7}{7!} Hagar, Pt.

Answer 17 There are various ways of answering this, you could calculate it directly using the formulas for $\cos(A+B)$ and $\sin(A+B)$. Similarly, we can define the principal branch of the logarithm $\textrm{Log}(z) = \log|z| + i\textrm{Arg}(z)$. To fill-in the historical period from the Exodus and Sinai revelation to the preparations in Moab to enter the Promised Land, B. So the modulus and argument of $e^{x} e^{i y}$ must be $r$ and $\theta$ respectively. Expanding $e^{i\theta}$ we get: So, what does this have to do with the polar coordinate notation? + \frac{z^4}{4!} \begin{eqnarray} 8\alpha^3 + 4\alpha^2 + 40\alpha + 20 &=& 8(-5\alpha) + 4(- 5)+40\alpha +20 \qquad &(\textrm{because} \quad \alpha^2 = -5 \quad \textrm{and} \quad \alpha^3 = -5 \alpha)& \\ &=& -40\alpha - 20 + 40\alpha + 20 \qquad &\textrm{(multiplying)}& \\ &=& 0 \qquad &\textrm{(simplifying)}& \end{eqnarray}. If $|z| = r = 1$ then $z = \cos\theta +i\sin\theta$, and so the formula in For example, the second-century BC Letter of Aristeas sees the behaviour of clean animals as models for human conduct. Since $x^y = e^{y\log(x)}$, this also means that $x^y$ is not unique when $x$ and $y$ are complex numbers. Find a power series expansion for $\cosh(z)$ and prove that $\cosh(i z) = \cos(z)$. branch of complex exponentiation to be $x^y = e^{y\textrm{Log}(x)}$. man too was vegetarian until after the flood.

The six laws about the land (22:50 to the end) similarly remind the reader that the promise is on the verge of fulfillment (Numbers, 15). The birds listed as unclean are unclean, because they are birds of prey, i.e. + \dots \\ &=& (1 - \frac{\theta^2}{2!}

Earlier commentators picked on certain elements in the food laws as suggestive of a particular interpretation. Two census are taken in Numbers (1; 26): a. The argument $\arg(z)$ of a complex number $z$ is the angle between the vector corresponding to $z$ and the positive $x$-axis. Exercise 15 What is $|z^n|$ in terms of $|z|$?

The Argand Plane

+ \dots$$ and so $$\cosh{i z} = 1 - \frac{z^2}{2!} Polar Coordinates and De Moivre's Formula Answer 10 The solution is, using the quadratic formula, $$ x = \frac { -2\pm i \sqrt{4-8}} {2}\ = 1\pm \sqrt{-1}\ = -1 \pm i $$ You can use wikipedia to look up any unfamiliar words or concepts. Can you guess what happens if you expand and simplify But precise dating of the material is largely irrelevant to exegesis, for it is the final form of the text that has canonical authority for the church ... (Numbers, 24-25). Oh yeah. [Hint: use De Moivre's formula and the fact that $\cos^2\theta + \sin^2\theta = 1$] This section is much more difficult, you need to be able to understand measuring angles in radians instead of degrees. 33:38; Deut. Exercise 2 Factorise the polynomial $x^2 - a^2$ (you don't need complex numbers for this) ), but that it is very unlike any number we've seen before. 9 For fuller discussions of this difficult matter see Wenham, Numbers, 60-66; Budd, Numbers, 6-9. $$ \frac {a^2 + b^2 }{a -i b}\ = \frac {(a^2 + b^2)(a + i b)}{a^2 + b^2}\ = a + i b $$ However, if we multiply the top and bottom by $1 - i$, we get: $$\frac{1}{1+i} = \frac{1- i}{(1+i)(1- i)} = \frac{1- i}{1+i- i+1} = \frac{1- i}{2} = \frac{1}{2} - i\frac{1}{2}$$ 23 Maryono writes, Moses also wanted Israel to learn from history. By induction, $q(z) = (z - w_1) \dots (z - w_n)$ and hence $p(z) = (z - w_1) \dots (z - w_n)(z - w_{n+1})$ and so the theorem is true by induction. Answer 23 We know $e^{i\pi} = - 1$ so $z^n = - 1$ means that $z^n = e^{i\pi}$. Well, using the formula above, $z^2 = (r^2 ,2\theta)$. It is not unreasonable to suppose that in addition to the written log of the stages of the journeyings (33:2) Moses also kept a record of the dates--at least those preserved in the account (OTS, 163, n. 1). If you've done any quadratic equations, you'll know that there is a nice formula for the solution of the quadratic equation $a x^2 + b x + c = 0$, given by: $$x = \frac { -b\pm \sqrt{b^2 - 4a c}} {2a}$$ 38:26), as is the case with the other Torah books. An Introduction to Complex Numbers. This makes it very easy to multiply and divide complex numbers written in polar coordinates, since $(r_1,\theta_1)(r_2, \theta_2) = (r_1 r_2 , \theta_1 + \theta_2)$. Exercise 26 [very difficult indeed] Find all complex number solutions to the equation $x^y = y^x$ .

The rondo, or variation, form in Exodus, Leviticus, and Numbers emphasizes large cycles which bring out “the parallels between the three journeys, and between the three occasions of law-giving, at Sinai, Kadesh and the plains of Moab.”18 The following charts emphasize this19, Egypt (Ex 1--13), Sinai (Ex 19--Num 10), Kadesh (Num 13-20), Plains of Moab (Num 22-36), A. This is called an eighth root of unity. So $|z w| = r_1 r_2$ and $\arg(z w) = \theta_1 + \theta_2$. What is $1/i$ for example?

Obedience to His commands will assure the possibility of enjoying the blessing in the land. Class Number Book Students will practice counting to ten and one to one association. Exercise 17 (very hard, for those who have done some trigonometry) What is $(\cos{\theta} +i\sin{\theta})^n$ in terms of $\cos{n\theta}$ and $\sin{n\theta}$? The other solutions are $- w$ , $i w$ and $- i w$. So, a solution to the equation $x^3 + x^2 + 20x + 20 = 0$ is $x = 2\alpha = \pm 2 i \sqrt{5}$. He argues that in Genesis 49 and Deuteronomy 22:110 the ox, the best of the sacrificial and clean animals, symbolizes Israel, while the ass, an unclean beast, pictures Canaan. c. Other non-literal approaches have been suggested for the reading of the numbers in the census: 1) The census totals are misplaced census lists from the time of David, 2) The census totals are part of the writer’s “epic prose” style intended to express the wholeness of Israel and the enormity of YHWH’s deliverance of the people (e.g., figurative), 3) The census totals are literary fiction and/or exaggerations corrupted by centuries of revising the Pentateuch, 4) The Hebrew word for “thousands” from the lack of vowel markings in the writings and could be read as “clan,” “tribe,” or even unit” (cf. Exercise 1 Work out $i^3 , i^4 , i^5 , i^6 , i^7$ and $i^8$. We also know that $\sin^2\theta = 1 - \cos^2\theta$ , Now we can discuss the amazing thing about the geometry of complex numbers. Also, $\cos(i z) = \cosh(z)$ and $\sin(i z) = i\sinh(z)$. So, if $b^2 - 4a c$ < 0, then $$ x = \frac { -b\pm i \sqrt{4a c - b^2}} {2a} $$ Amazingly, the formula also works if $a$, $b$ and $c$ are complex numbers! We won't be able to prove the second equation until after the next section, but we can prove the first one.

A decimal is any number, including whole numbers, in our base-ten number system. + \dots \\ &=& 1 + i\theta - \frac{\theta^2}{2!} An example of his discussion of an anthropologically-based approach to ritual symbolism is as follows: First, this approach seeks to understand the whole ritual system and not just parts of it, or more precisely to understand the parts in the light of the whole. Suppose $z = x+i y$ and $w = (1, \phi) = \cos\phi + 7 This following is adapted form Hill and Walton, SOT, 136. Elsewhere the text implies that priests were also recording and preserving the divine instruction and regulations, especially those pertinent to their duties associated with the tabernacle (cf. I've tried to make the exercises less like the standard computational ones you get at school, but this means that some of them are quite hard. Therefore, the combination of both the real number and imaginary number is a complex number.. 14 For a good discussion of the literary structure of particular units in Numbers see Migrom, Leviticus, xii-xxxi.

Answer 3 $ \quad (x- i a)(x+i a)$. If you don't know about how to add two vectors, look at the following picture: So, adding vectors corresponds to adding complex numbers. Answer 25 We're trying to prove it by induction, so we have to first prove that it is true for $n = 1$. [Hint: $z = (z/w)(w)$] Then Moses warns the people ... [that] grave consequences shall [occur if] they fail to obey God. So, if $|z| = r_1, |w| = r_2$, $\arg(z) = \theta_1$ and $\arg(w) = \theta_2$ then $z = r_1 (\cos\theta_1 +i\sin\theta_1)$ and $w = = r_2 (\cos\theta_2 +i\sin\theta_2)$. The Argand Plane The other $n$ solutions are $z^2$, $z^3$,\dots, $z^n = 1$. To answer this, we introduce something called the complex conjugate . What is $\arg(z/w)$ in terms of $\arg(z)$ and $\arg(w)$?

If you know about mathematical induction, prove your result. 22 Budd writes, One of his [the author's] chief concerns is to establish principles of attitude and behavior which are a precondition of possession and enjoyment of the land (Numbers, xvii). 17 The clearest example of this is to be found in chapter 15, where the demand to offer grain, oil, and wine along with animal sacrifice is an implicit pledge that one day Israel will enter Canaan despite the events described in the previous chapters 13--14. You might not have seen radians used before, they're just another way of measuring angles.
Expand and simplify $(\cos(2\pi /4 )+ i\sin(2\pi /4 ))^4$.

If we write $z = (r,\theta)$ then we can also write $z = {r}{e^{i\theta}}$ using the proof above. + \dots$$ which is just the power series for $\cos(z)$. If you've ever done vectors, this will look very familiar, a 2D vector can be written $a\mathbf{i} +b\mathbf{j}$ where $\mathbf{i}$ and $\mathbf{j}$ are the unit vectors. you to test your understanding, with answers at the back.

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