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# How to multiply numbers - Part 1 Complex numbers and multiplication.
The ancient mathematicians needed to do a lot of manual calculations, such as multiplication. But without a calculator or computer, how can they do it fast? For examlpe, what is $8472839759245687346\times 98572398897898972343$?
If one is only interested in an estimate, perhaps one can trade accuracy for speed, by consulting some precomputed table of values. This led to two approaches, one using trigonometric identities(!) and another using logarithms.
We sill start with the trigonometric approach first. For this, we will need trigonometric identities. But instead of looking them up or deriving them using elementary geometry, we will take a detour approach using **complex numbers!**
We will investigate what does it mean geometrically when you add complex numbers together, and when you multiply complex numbers together.
## Complex numbers.
A complex number $z$ is a quantity of the form $z=a+ib$ where $a,b$ are both real numbers, and $i$ is the **imaginary unit** such that $i^2=-1$. We say $a$ is the **real part** of $z=a+ib$, and write $a=\mathfrak{Re}(z)$, and $b$ is the **imaginary part** of $b$, and write $b=\mathfrak{Im}(z)$. Using this rule of $i^2=-1$, you can carry out addition, multiplication by using the rules of distribution, and grouping terms with $i$, and terms without:
$$
\begin{align}
(a+ib)+(c+id) & =(a+c)+i(b+d) \\
(a+ib)(c+id) & =(ac-bd)+i(bc+ad)
\end{align}
$$
Given a complex number $z=a+ib$, its **complex conjugate** is $\bar{z}=a-ib$.
![[---images/---assets/---icons/question-icon.svg]] Some warm-up. Let $z=1+2i$ and $w=4+3i$. Find
(1) $z+w$
(2) $zw$
(3) $z\bar{z}$
(4) What is $i^2$, $i^3$, $i^4$, $i^5$, $i^6$? What do you observe? What is $i^{137}$?
We can plot a complex number $z=a+ib$ on the complex plane, taking the **Cartesian coordinate** $(a,b)$, where $a$ is the real part of $z$ and $b$ is the imaginary part of $z$ :
![[1 teaching/summer program 2023/week 3/---files/How-to-multiply-numbers-1 2023-05-24 11.15.22.excalidraw.svg]]
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Another way to describe the complex number $z$ on the complex plane is to use **polar coordinates**. The length $r$ of the line segment from $0$ to $z$ is the **magnitude** of $z$, sometimes we write it as $|z|$. Note $|z|=\sqrt{ a^2+b^2 }$ by Pythagorean theorem. The **angle** $\theta$ formed by this line segment with the positive $x$-axis is called the **argument** of $z$, or $\arg(z)$:
![[1 teaching/summer program 2023/week 3/---files/How-to-multiply-numbers-1 2023-05-23 16.18.01.excalidraw.svg]]
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![[---images/---assets/---icons/question-icon.svg]] Express $a$ and $b$ in terms of $r$ and $\theta$ in above diagram. Yep, trigonometry time!
This shows that every complex number $z$ can be expressed in **polar form**, where $z=r(\cos \theta + i\sin \theta)$, for some nonnegative $r\geq 0$ that represents the magnitude of $z$. In other words, the **polar coordinates** of the complex number $z$ is $(r,\theta)_{\text{{pol}}}$.
![[---images/---assets/---icons/question-icon.svg]] If $z=1+2i$ and $w=4+3i$, draw out $z$, $w$, and $z+w$ on the complex plane. What kind of shape do the points $0$, $z$, $w$, and $z+w$ form?
![[1 teaching/summer program 2023/week 3/---files/How-to-multiply-numbers-1 2023-05-24 15.12.04.excalidraw.svg]]
![[---images/---assets/---icons/question-icon.svg]] If $z=a+ib$, which has a coordinate of $(a,b)$ on the complex plane, and $w=c+id$ has Cartesian coordinate of $(c,d)$ on the complex plane. What is the coordinate of $z+w$? If you treat these as **vectors** (an arrow with magnitude), can you describe the location of $z+w$ in terms of the vectors described by $z$ and $w$?
![[---images/---assets/---icons/question-icon.svg]] If $z=1+2i$, what is $4z$? Draw both $z$ and $4z$ on the complex plane below. Can you describe what is happening to the vector described by $z$ when we multiply it by 4?
![[1 teaching/summer program 2023/week 3/---files/How-to-multiply-numbers-1 2023-05-24 15.12.04.excalidraw.svg]]
![[---images/---assets/---icons/question-icon.svg]] If $z=1+2i$, what is $zi$? Draw both $z$ and $zi$ on the complex plane below. Can you describe what is happening to the vector described by $z$ when we multiply it by $i$?
![[1 teaching/summer program 2023/week 3/---files/How-to-multiply-numbers-1 2023-05-24 15.12.04.excalidraw.svg]]
![[---images/---assets/---icons/question-icon.svg]] If $z=1+2i$ and $w=4+3i$, what is $zw$? Draw out $z$, $w$, and $zw$ on the complex plane below. Can you make a guess what is happening to the vector described by $z$ when we multiply it by $w$? Hint: Maybe examine the arguments of all these complex numbers.
![[1 teaching/summer program 2023/week 3/---files/How-to-multiply-numbers-1 2023-05-24 15.12.04.excalidraw.svg]]
As it turns out when we have two complex numbers $z=a+ib$ and $w=c+id$, their product $zw=(ac-bd)+i(ad+bc)$ is a new complex number whose magnitude is $|z| |w|$ and argument is $\arg(z)+\arg(w)$. So the **magnitudes multiply together**, and the **angles add together**! ![[---images/---assets/---icons/exclaim-icon.svg]]
In polar form, this means if we have $z=r(\cos \alpha+i\sin \alpha)$ and $w=s(\cos \beta+i\sin \beta)$, then the product
$$
zw=rs(\cos (\alpha+\beta)+i\sin(\alpha+\beta))
$$
If we symbolize everything in polar coordinates, then $\underbrace{(r,\alpha)_\text{pol}}_z \cdot\underbrace{(s,\beta)_{\text{pol}}}_w=\underbrace{(rs,\alpha+\beta)_{\text{pol}}}_{zw}$.
Geometrically:
![[1 teaching/summer program 2023/week 3/---files/How-to-multiply-numbers-1 2023-05-24 09.41.05.excalidraw.svg]]
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Let us see why!
## A hands-on example first.
![[---images/---assets/---icons/question-icon.svg]] Let us dissect what happens if we multiply two complex numbers $zw$ together. We will step-by-step multiply $z=1+2i$ with $w=4+3i$ by distributing $z$ to each parts in $w$, and plotting the results of each step in the complex plane below. Let us see what happens
(1) Plot $z=1+2i$ and $w=4+3i$ on the complex plane below. Label them $z$ and $w$
Now, since $w=4+3i$, we have $zw=z(4+3i)=4z+3iz$ by distribution. Let us figure out what what each part is.
(2) Calculate and plot $4z$ in the complex plane below.
(3) Calculate and plot $iz$ in the complex plane below. What is the angle between $z$ and $iz$?
(4) Calculate and plot $3iz$ in the complex plane below.
(5) Plot $4z+3iz$ in the complex plane below. This is now the product $zw$ that we seek!
(6) Draw the box formed by $0,4z,3zi,4z+3iz$, and draw a box whose diagonal is $w$ with sides parallel to the axes. What can you say about these two boxes?
(7) What can you say about the angle of $zw$ ? And what can you say about its magnitude? Express them in terms of angles and magnitudes of $z$ and $w$.
![[1 teaching/summer program 2023/week 3/---files/How-to-multiply-numbers-1 2023-05-24 15.12.04.excalidraw.svg]]
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///to check///
If you do it right, you should get something like this
![[1 teaching/summer program 2023/week 3/---files/How-to-multiply-numbers-1 2023-05-24 16.09.31.excalidraw.svg]]
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///
## Analysis of complex number multiplication.
Let us say $z=a+ib$ and has polar form $z=r(\cos \alpha+i\sin \alpha)$, and $w=c+id$ with polar form $w=s(\cos \beta+i\sin \beta)$.
We will breakdown the product $zw=z(c+id)=z(c)+z(i)(d)$ step-by-step as how it is distributed.
![[---images/---assets/---icons/question-icon.svg]] The complex number $z=a+ib$ is sketched below, calculate $zc$ and describe geometrically what happens if we multiply $z$ by a real number $c$. If we shade in the box with diagonal formed by $z$, what does the box $zc$ look like? What is the magnitude of $zc$ in terms of $r$ ? What is the Cartesian coordinate of $zc$, and what is its polar coordinate?
![[1 teaching/summer program 2023/week 3/---files/How-to-multiply-numbers-1 2023-05-23 17.12.28.excalidraw.svg]]
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![[---images/---assets/---icons/question-icon.svg]] The complex number $z=a+ib$ is sketched below, calculate $zi$ and describe geometrically what is the effect of multiplying $z$ by $i$ ? How does the box formed by $zi$ related to the box formed by $z$? What is the magnitude of $zi$ in terms of $r$? What is the Cartesian coordinate and polar coordinate of $zi$?
![[1 teaching/summer program 2023/week 3/---files/How-to-multiply-numbers-1 2023-05-23 17.24.42.excalidraw.svg]]
%%[[1 teaching/summer program 2023/week 3/---files/How-to-multiply-numbers-1 2023-05-23 17.24.42.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/week 3/---files/How-to-multiply-numbers-1 2023-05-23 17.24.42.excalidraw.dark.svg|dark exported image]]%%
![[---images/---assets/---icons/question-icon.svg]] Continuing with $zi$ we have before, and now multiply by real number $d$ to get $zid$, what is the effect of multiplying by $d$ here? What is magnitude of $zid$ in terms of $r$? What is the Cartesian coordinate and polar coordinate of $zid$?
![[1 teaching/summer program 2023/week 3/---files/How-to-multiply-numbers-1 2023-05-23 17.29.48.excalidraw.svg]]
%%[[1 teaching/summer program 2023/week 3/---files/How-to-multiply-numbers-1 2023-05-23 17.29.48.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/week 3/---files/How-to-multiply-numbers-1 2023-05-23 17.29.48.excalidraw.dark.svg|dark exported image]]%%
![[---images/---assets/---icons/question-icon.svg]] Finally, let us add them together, that $z(c+id)=zc+zid$. This is the product $zw$ that we seek. Here we get a rectangular box formed $0$, $zc$, $zid$, and $zc+zid$. Why is it rectangular? What are the side lengths of this box? What is the diagonal of this box, equivalently, the magnitude of $zw$? Now note the angle of $zw$ is $\alpha + ???$, which we would like to figure out what $???$ is... (next problem)
![[1 teaching/summer program 2023/week 3/---files/How-to-multiply-numbers-1 2023-05-23 17.41.51.excalidraw.svg]]
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![[---images/---assets/---icons/question-icon.svg]] If we rotate the big box formed by $zc+zid$ above, and compare it with the box formed by $w=c+id=s(\cos \beta+i\sin \beta)$, what can you conclude about the angle of $zc+zid$ above? Why can we make this conclusion?
![[1 teaching/summer program 2023/week 3/---files/How-to-multiply-numbers-1 2023-05-23 17.52.09.excalidraw.svg]]
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![[---images/---assets/---icons/question-icon.svg]] Finally, can you conclude what we can say about the product $zw$ if $z=r(\cos \alpha +i\sin \alpha)$ and $w=s(\cos \beta+i\sin \beta)$?
**Hurray!! You proved one of the most important properties of complex numbers ! ! ! GOOD JOB ! ! !** Let us reiterate: If $z=a+ib$ has polar form $z=r(\cos \alpha+i\sin \alpha)$, and $w=c+id$ has polar form $w=s(\cos \beta+i\sin \beta)$, then their product $zw=(ac-bd)+i(bd+ad)$ has polar form $zw=rs(\cos(\alpha+\beta)+i\sin(\alpha+\beta))$. That is, **the magnitudes multiply**, and the **angles add**!