How to of the Day

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• How to Cook Sweet Potatoes
• How to Remove Calluses Naturally
• How to Understand Complex Numbers

How to Cook Sweet Potatoes Posted: 01 Feb 2017 04:00 PM PST Sweet potatoes are a healthy, sweet alternative to regular potatoes. This wikiHow will show you several ways of cooking them. Prep time: 20 minutes Cook time: 45-60 minutes Total time: 65-80 minutes EditKey Points Bake them in the oven at 400 °F (204 °C) for 45 - 60 minutes. More ↓ Microwave them on HIGH, uncovered, for 8 to 10 minutes. ↓ Cook them in a slow cooker on LOW for 4 to 6 hours in 1/4 to 1/2 (60 to 120 milliliters) of water. ↓ Boil peeled and cubed sweet potatoes for 12 minutes. Drain the water, add some butter, then mash them. ↓ EditIngredients EditOven Baked Sweet Potatoes[1] 8 unpeeled medium sweet potatoes 8 tbsp. of butter Salt to taste Pepper to taste EditMicrowave Baked Sweet Potatoes[2] 8 unpeeled medium sweet potatoes 8 tbsp. of butter Salt to taste Pepper to taste EditSlow Cooked Sweet Potatoes 8 unpeeled medium sweet potatoes 6 tbsp. of butter 1/4 to 1/2 cup (60 to 120 ml) water Salt to taste Pepper to taste EditMashed Sweet Potatoes[3] 8 peeled medium sweet potatoes 1/4 to 1/2 cup (60 to 125 ml) butter Salt to taste Pepper to taste 1/3 cup (80 ml) sour cream 1/4 cup (60 ml) milk EditSteps EditMaking Oven Baked Sweet Potatoes Preheat the oven to . Line a rimmed baking sheet with nonstick aluminum foil. Scrub the skin of the potatoes clean. Use a vegetable brush and running water to remove any dirt and grime off the skin of the sweet potatoes. Pierce the exterior of each sweet potato. Use the tines of a fork to make holes in the skin of each potato, piercing the potato three or four times. Place the pierced sweet potatoes on the prepared baking sheet. Bake the uncovered sweet potatoes in the oven. The sweet potatoes need to bake until they become tender, which will usually take between 45 - 60 minutes. Soften the flesh. Place each sweet potato inside a clean kitchen towel. Roll the potato on the counter, pressing on it gently. Doing so loosens the flesh inside. Cut open each sweet potato. Use a knife to slit open each potato from one end to the other. Serve the sweet potatoes hot with butter, salt, and pepper. Each sweet potato should be topped with about 1 tbsp. (15 ml) of butter. Add salt and pepper to taste. EditMaking Microwave Baked Sweet Potatoes Scrub the sweet potatoes clean. Wash the exterior of each sweet potato under running water, using a vegetable brush to remove any stubborn dirt. Pierce the skin. Poke into each sweet potato three to five times with the tines of a fork. Place the sweet potatoes in a microwave-safe dish. You can also use a microwave-safe plate with a rim. Do not cover. Microwave the sweet potatoes until tender. Set the microwave on full power and cook for 8 to 10 minutes. Pause the microwave halfway in between to flip and rotate the sweet potatoes to ensure even cooking. Slice open each sweet potato. Cut an "X" in the top of each potato and gently press on the sides of the skin to push the flesh upward. Serve with butter. Use about 1 Tbsp. (15 ml) of butter per sweet potato. Add salt and pepper to taste. EditMaking Slow Cooked Sweet Potatoes Wash the sweet potatoes. Scrub the skin with a vegetable brush under running water. Poke holes in the skin. Use the tines of a fork to pierce the outside of each sweet potato several times, covering the exterior in tiny holes. Place the sweet potatoes into a 5 or 6 quart (5 or 6 liter) slow cooker. The sweet potatoes can be piled up near the top, but you should still be able to cover the slow cooker with a lid without any difficulty. Add 1/4 to 1/2 cup (60 to 120 ml) water to the slow cooker. A little water will help the sweet potatoes cook better, but there should not be enough water to cover them. You do not want them to boil in the slow cooker. Instead, the water is there to prevent the sweet potatoes from drying out and potentially burning. Cook the sweet potatoes for 4 to 6 hours. Cover the slow cooker and set the heat to low, cooking the sweet potatoes until they become tender. Slice open and serve. Cut the sweet potatoes open with a knife or fork. They should be tender enough to fall apart without needing to loosen the flesh using other techniques. Add 6 tbsp. of butter and salt and pepper to taste before serving. EditMaking Mashed Sweet Potatoes Peel and cube the sweet potatoes. Use a vegetable peeler to remove the skin from each sweet potato. Cut each one into 1/2-in. (1.27-cm) cubes. Place these cubes in a colander and rinse with running water to remove any traces of dirt. Transfer the sweet potato cubes into a 4 quart (4 liter) pot. Fill the pot with enough water to cover the cubes. Add a dash of salt to the water. Boil the sweet potatoes for 12 minutes. You can add a pinch of salt to the pot if you like. Cover the pot and heat the water and cubed sweet potatoes over medium-high to high heat, until they feel tender when pierced with a fork. Drain the water. Pour the contents of the pot through a colander. Separate the water out and return the potatoes to the pot. Add the butter to the sweet potatoes. You can add anywhere from 1/4 to 1/2 cup (60 to 125 ml) of butter, depending on how creamy you want the mashed sweet potatoes to be. Allow the butter to sit on the hot sweet potatoes until it melts, stirring as necessary to speed up the melting process. Mash the sweet potatoes with a potato masher. Smash the sweet potatoes and butter together using the masher until the mixture becomes smooth. If you do not have a potato masher, you can also use a hand-held electric mixer. Add the remaining ingredients. Stir in 1/3 cup (80 ml) of sour cream and 1/4 cup (60 ml) of milk and add salt and pepper to taste. Continue stirring with a large spoon or fork until the mixture is creamy and thoroughly combined. Return the pot to the oven. Cook the mashed sweet potatoes over low heat, stirring frequently, until the sweet potatoes are heated through. Then, serve them while they're hot. EditVideo EditTips Instead of flavoring your sweet potatoes with butter, salt, and pepper, you can also flavor them with cinnamon, brown sugar, paprika, and numerous other sweet and spicy seasonings. The seasoning you choose should depend on your own sense of taste. Aside from these basic ways to cook sweet potatoes, there are also numerous other ways to prepare them. You can slice them and bake them with a glaze, cut them into wedges and bake them as fries, or you can puree them and use them for baked goods like bread, pies, and cakes. Store your sweet potatoes in a cool, dark place before ready to use. Sweet potatoes decay and grow moldy rather easily, especially if stored poorly. Do not keep the sweet potatoes in the refrigerator, since refrigeration can cause sweet potatoes to lose their flavor. EditThings You'll Need Vegetable brush Baking sheet Nonstick aluminum foil Microwave-safe dish Fork Knife 5 or 6 quart (5 or 6 liter) slow cooker Vegetable peeler 4 quart (4 liter) pot Potato masher or hand-held electric mixer EditRelated wikiHows Cook a Sweet Potato in the Microwave Use-Up-Left-over-Bolognese-Sauce-or-Chilli EditSources and Citations Videos provided by Fablunch Cite error: <ref> tags exist, but no <references/> tag was found

How to Remove Calluses Naturally Posted: 01 Feb 2017 08:00 AM PST Calluses are areas of hardened skin that usually develop on weight-bearing areas. Most calluses are on the feet and are caused by poorly fitting shoes or not wearing socks. The pressure of badly fitting shoes and the friction of going sockless can cause the skin to react, resulting in corns and calluses. On the hands, the most common causes are playing musical instruments or using any sort of hand tool--even a pen-- that causes pressure and friction. [1] For a healthy person, calluses can usually be treated at home using methods that involve softening the skin and rubbing away the callus. EditSteps EditRecognizing Calluses Recognize what a callus looks like. A callus is a small spot of hardened, thickened skin that develops from pressure or friction. Most often, they occur on the soles of the feet or on the hands or fingers.[2] Calluses are not contagious, but they can become uncomfortable if they get too large.[3] Know the difference between a callus and a corn. Corns and calluses are terms that are often used interchangeably. They do share some similarities, but there are also some differences. Technically, corns are areas of hardened skin near a bony area. Corns are usually found on or between the toes. Calluses are not associated with bony areas and usually appear on weight-bearing areas. Both corns and calluses are caused by friction, such as the foot rubbing on shoes or toes rubbing against each other. [4] Another difference between corns and calluses is that the callus is all thickened skin, but a corn has a hard central core, surrounded by reddened and inflamed tissue. Corns tend to be painful, while calluses are rarely painful. Contact your physician if your callus is painful. If your callus becomes infected, inflamed or painful, you should talk to your doctor. It may require medical treatment. EditSoftening the Skin Soak the callus in hot water. The simplest thing one can do is to soak the feet in hot water. Take a medium-sized tub and fill it up with warm water, around 45°C (113°F) and while sitting on a chair or stool, immerse your feet in the tub for 15 to 20 minutes while you relax and read a book. Add Epsom salts for additional skin softening.[5] Soften the hardened skin by adding a half cup of Epsom salts for every gallon of water. Soak the affected area for 10 to 20 minutes. At the end of this foot spa of sorts, you will find your calluses getting soft. Over a few days of repeating this procedure they will be soft enough for you to just scrape them off with your hand.[6] Massage castor oil into the callus. Castor oil can be used to soften skin and encourage new skin growth. Apply castor oil by massaging it into the callus. Then cover your feet or hands with old cotton socks or old cotton gloves. Castor oil will stain clothing, so be sure to choose something that can get stained. Cotton clothing is preferred because cotton is a natural fiber that will absorb the castor oil, but it will also leave some castor oil on the callus. Leave on for at least 30 minutes. Cover the callus in Vitamin E. Take a caplet of Vitamin E containing 400IU and use a needle to poke a hole in the caplet. Squeeze out the Vitamin E and massage it into the callus. Use as many caplets as needed to cover the entire callus. Leave the Vitamin E on the callus for at least 30 minutes. Make an aspirin paste. Aspirin contains salicylic acid, which helps treat calluses. Make a paste by crushing six uncoated aspirin tablets in a bowl. Add a half-teaspoon of either apple cider vinegar or lemon juice to make a paste. Apply the mixture onto the callus. Wrap the affected area in a warm towel and leave it on for 10 to 15 minutes.[7] EditUsing a Pumice Stone Purchase a pumice stone. Pumice is an extremely porous rock that forms during volcanic eruptions. [8] It can be used to gently rub off (exfoliate) the thickened skin of a callus. Once the callused area is softened, you'll use a pumice stone to rub off the top layers of the callus. Pumice stones are readily available at drug stores and grocery stores. Moisturize the callused area. Use one of the softening methods to get your callused area ready. Moisturize by leaving on castor oil or Vitamin E for at least 30 minutes, or you can leave on one of these treatments overnight. Rub the pumice on the callused area. Use the pumice after the moisturizing step to make it easier to gently remove the thickened skin. When your skin is softened, you won't need to rub too hard. Use gentle, firm, single direction strokes, such as those similar to how you would file your nails or play a violin. With a steady hand and constant, minimal pressure, rub off the top layer of the calluses to bring about healthy skin from underneath. Always remember that the callus is your body's response to increased pressure and friction. Rubbing too hard may result in more callus formation. Repeat this process every day. Be patient with the process of removing the callus. Use the pumice every day to remove a fraction of the callus. This may take a while, but will pay off in the long run. Talk to your doctor if the callus won't go away. If after one to two weeks, the callus is still present, call your physician for advice. The callus may require a medical procedure such as:[9] Surgical trimming The use of urea (a cleaning agent that helps loosen skin[10]) to soften and remove the skin cells Orthotics to minimize pressure and/or friction More extensive surgery Don't try to cut or shave the callus. Although the skin on a callus is hardened, you should remove the skin by rubbing it away. Do not try cutting or shaving the area. This can cause infections and laceration injuries. You can easily cut too deeply or at the wrong angle. You may require medical attention if you do this.[11] EditPreventing Calluses from Forming Examine your skin regularly for calluses. Monitor your skin for changes that might indicate a callus is forming. If you can't reach or see your feet, ask someone to help you. You can visit a doctor or podiatrist for a foot examination.[12] Stop the activity that's causing the callus. If you are developing a callus from playing a guitar, for example, you can stop doing that activity. It may not be feasible to stop the activity, however. For example, if you have a callus on your finger from using a pen for writing, you may not be able to eliminate this activity from your daily routine. Get shoes that fit properly. Many people develop calluses on their feet when their shoes don't fit. Since calluses are the skin's response to pressure or friction, you need to remove the source of the pressure or friction. Get your feet measured. As you age, your feet will change size and shape, so it's important to keep your shoe size current.[13] Try on shoes before you buy them. Sometimes, the fit will be different depending on the manufacturer, so pay attention to how the shoe feels on your foot, not the size marked on the box. Make sure there is about ½ inch between your toe and the tip of the shoe. Don't buy shoes expecting them to stretch as you wear them. If they are too tight when you buy them, then go up a size. Protect your skin from calluses. Wear gloves, socks and properly fitting shoes to protect your skin from calluses. Don't walk around barefoot, as this increases the potential for callus formation.[14] Use moisturizing foot and hand creams. Using these lotions on your feet and hands before you wear your shoes or gloves to reduce friction can greatly alleviate your callus pain. Alternatively, consider slathering them in petroleum jelly. Moisture will never be a problem again! Use orthopedic shoe inserts. These or the donut shaped footpads specifically recommended for calluses are fantastic as they keep the callused area raised and cushioned, hence reducing friction by avoiding contact with footwear. They won't get rid of existing calluses, but they will prevent new ones from forming.[15] You can also make a foot pad out of moleskins by cutting two moon-shaped pieces and arranging them around your callus. EditWarnings For people with diabetes or circulatory disorders, both corns and calluses can be significant problems. If you have diabetes or a circulatory disorder, talk to your physician before trying to remove a callus. Even minor cuts or injuries can cause potentially serious consequences such as foot ulcers. EditRelated wikiHows Remove Corns from Your Toes Remove Dry Skin from Your Feet Using Epsom Salt Remove Warts Naturally Using Garlic Heal Cracked Skin Exfoliate Feet for a Pedicure EditSources and Citations Cite error: <ref> tags exist, but no <references/> tag was found

How to Understand Complex Numbers Posted: 01 Feb 2017 12:00 AM PST When we first learned to count, we started with the natural numbers – 1, 2, 3, and so on. Soon after, we added 0 to represent the idea of nothingness. Then, we added the negative numbers to form the integers, which were slightly less intuitive, but concepts like debt helped solidify our grasp of them. The numbers that filled in the gaps between the integers consist of the rational numbers – numbers that can be written in terms of a quotient of two integers ${\displaystyle {\frac {a}{b}}}$ – and the irrational numbers, which cannot. Together, these numbers make up the field called the real numbers. In mathematics, this field is commonly denoted by ${\displaystyle {\mathbb {R} }.}$ However, there are many applications where real numbers fail to solve problems. One of the simplest examples is the solution to the equation ${\displaystyle x^{2}+1=0.}$ There exist no real solutions, but according to the fundamental theorem of algebra, there must be two solutions to this equation. In order to accompany those two solutions, we need to introduce the complex numbers ${\displaystyle {\mathbb {C} }.}$ This article aims to give the reader an intuitive understanding of what complex numbers are and how they work, starting from the bottom up. EditSteps EditDefinition of a Complex Number Define complex number. A complex number is a number that can be written in the form ${\displaystyle a+bi,}$ where ${\displaystyle i={\sqrt {-1}}.}$ The most important part of this number is what ${\displaystyle i}$ is. It is not found on the real number line at all. Some examples of complex numbers are listed below. Notice that the number 3 is a complex number. It just has an imaginary component equal to 0, because ${\displaystyle 0i=0.}$ ${\displaystyle 2+3i}$ ${\displaystyle 4-i}$ ${\displaystyle 3}$ By convention, complex numbers are denoted using the variables ${\displaystyle z}$ and ${\displaystyle w,}$ similar to ${\displaystyle x}$ and ${\displaystyle y}$ denoting some real numbers. So we say that ${\displaystyle z=a+bi.}$ Some authors may say ${\displaystyle z=x+iy.}$ As we can see, now we have a solution to the equation ${\displaystyle x^{2}+1=0.}$ After using the quadratic formula, we have ${\displaystyle x=\pm i.}$ Understand the powers of ${\displaystyle i}$. We said that ${\displaystyle i={\sqrt {-1}}.}$ Then ${\displaystyle i^{2}=-1.}$ If we multiply that with ${\displaystyle i}$ again, we get ${\displaystyle i^{3}=-i.}$ Multiply ${\displaystyle i^{2}}$ with itself and we get ${\displaystyle i^{4}=1.}$ This underscores a strange property of the imaginary unit. It takes four cycles to get to 1 (a positive number), whereas a number on the real number line -1 takes just two. Differentiate between real numbers and purely imaginary numbers. A real number is a number that you are already familiar with; it exists on the real number line. A purely imaginary number is a number that is some multiple of ${\displaystyle i.}$ The key concept to note here is that none of these purely imaginary numbers lie on the real number line. Instead, they lie on the imaginary number line. Below are some examples of real numbers. ${\displaystyle 6}$ ${\displaystyle {\frac {145}{231}}}$ ${\displaystyle \pi }$ Below are some examples of imaginary numbers. ${\displaystyle 2i}$ ${\displaystyle ({\sqrt {3}}+{\sqrt {5}})i}$ What do all five of these numbers have in common? They are all part of the field known as the complex numbers. The number 0 is notable for being both real and imaginary. Extend the real number line to the second dimension. In order to facilitate the imaginary numbers, we must draw a separate axis. This vertical axis is called the imaginary axis, denoted by the ${\displaystyle {\mathrm {Im} }}$ in the graph above. Similarly, the real number line that you are familiar with is the horizontal line, denoted by ${\displaystyle {\mathrm {Re} }.}$ Our real number line has now been extended into the two-dimensional complex plane, sometimes called an Argand diagram. As we can see, the number ${\displaystyle a+bi}$ can be represented on the complex plane by drawing an arrow from the origin to that point. A complex number can also be thought of as the coordinates on a plane, though it is extremely important to understand that we are not dealing with the real xy-plane. It just looks the same because both are two-dimensional. Perhaps one of the most nonintuitive part of understanding complex numbers is that every number system that we have dealt with – integers, rationals, reals – are deemed to be "ordered." For example, it makes sense to think of 6 as being greater than 4. But in the complex plane, it is meaningless to compare if ${\displaystyle 4+i}$ is greater than ${\displaystyle 3+2i.}$ In other words, the complex numbers are an unordered field. Break the complex numbers up into the real and imaginary components. By definition, every complex number can be written in the form ${\displaystyle z=a+bi.}$ We know that ${\displaystyle i={\sqrt {-1}},}$ so what do ${\displaystyle a}$ and ${\displaystyle b}$ represent? ${\displaystyle a}$ is called the real part of the complex number. We denote this by saying that ${\displaystyle \operatorname {Re} z=a.}$ ${\displaystyle b}$ is called the imaginary part of the complex number. We denote this by saying that ${\displaystyle \operatorname {Im} z=b.}$ (Important!) Both the real and imaginary parts are real numbers. So when someone refers to the imaginary part of some complex number ${\displaystyle w=c+di,}$ they always refer to the real number ${\displaystyle d,}$ not ${\displaystyle di.}$ Certainly, ${\displaystyle di}$ is an imaginary number. But it is not the imaginary part of the complex number ${\displaystyle w.}$ As a basic exercise, find the real and imaginary parts of the complex numbers given in step 1 of this part. Define the complex conjugate. The complex conjugate ${\displaystyle {\bar {z}}=a-bi}$ is defined as ${\displaystyle z}$ but with the sign of the imaginary part reversed. Conjugates are very useful in a number of scenarios. You may already be familiar with the fact that complex solutions to polynomial equations come in conjugate pairs. That is, if ${\displaystyle z_{0}}$ is a solution, then ${\displaystyle {\bar {z_{0}}}}$ must also be one as well. What is the significance of conjugates on the complex plane? They are the reflection over the real axis. As seen in the diagram above, the complex number ${\displaystyle z=x+iy}$ has a real part ${\displaystyle x}$ and an imaginary part ${\displaystyle y.}$ Its conjugate ${\displaystyle {\bar {z}}=x-iy}$ has the same real part ${\displaystyle x}$ but a negated imaginary part ${\displaystyle -y.}$ Think of complex numbers as a collection of two real numbers. Because complex numbers are defined such that they consist of two components, it makes sense to them of them as two-dimensional. From this perspective, it makes more sense to make analogies using functions of two real variables, instead of just one, even though most complex functions are functions of one complex variable. EditArithmetic Extend the methods of arithmetic to complex numbers. Now that we know what complex numbers are all about, let's do some arithmetic with them. Complex numbers are similar to vectors in this sense, because we add and subtract their components. Let's say that we wanted to add two complex numbers ${\displaystyle z=a+bi}$ and ${\displaystyle w=c+di.}$ Then adding these two complex numbers is as simple as adding the real and imaginary components separately. All we do is to add the real parts, add the imaginary parts, and sum them up. ${\displaystyle z+w=(a+c)+(b+d)i}$ The same idea works for subtraction as well. ${\displaystyle z-w=(a-c)+(b-d)i}$ Multiplication is similar to FOILing from algebra. ${\displaystyle zw=(a+bi)(c+di)=(ac-bd)+(bc+ad)i}$ Division is similar to rationalizing the denominator from algebra as well. We multiply the numerator and the denominator by the conjugate of the denominator. ${\displaystyle {\frac {z}{w}}={\frac {a+bi}{c+di}}={\frac {(a+bi)(c-di)}{(c+di)(c-di)}}={\frac {ac+bd}{c^{2}+d^{2}}}+{\frac {bc-ad}{c^{2}+d^{2}}}i}$ The point of showing these steps is not to derive formulas to memorize, even though they do work. The point is to show that the operations of addition, subtraction, multiplication, and division of two complex numbers all must output another complex number that can be written in the form ${\displaystyle z=x+iy.}$ Adding two complex numbers gives another complex number, dividing two complex numbers also gives another complex number, etc. While messy, the above substeps were shown so that we are confident that the arithmetic of complex numbers is consistent with the way that we have defined them. Extend the addition properties of real numbers to complex numbers. You are familiar with the commutative and associative properties of real numbers. Such properties extend into the complex numbers as well. Adding two complex numbers is commutative, because we are adding the real components separately, and we know that addition of real numbers is commutative. ${\displaystyle z+w=w+z}$ Adding two complex numbers is associative, for a similar reason. ${\displaystyle (z+w)+v=z+(w+v)}$ There exists an additive identity of the complex number system. This identity is called 0. ${\displaystyle z+0=z}$ There exists an additive inverse of a complex number. The sum of a complex number with its additive inverse is 0. ${\displaystyle z+(-z)=0}$ Extend the multiplication properties of real numbers to complex numbers. The commutative property holds for multiplication. ${\displaystyle zw=wz}$ The associative property holds for multiplication as well. ${\displaystyle (zw)v=z(wv)}$ The distributive property holds for complex numbers. ${\displaystyle z(w+v)=zw+zv}$ There exists a multiplicative identity of the complex number system. This identity is called 1. ${\displaystyle z(1)=z}$ There exists an multiplicative inverse of a complex number. The product of a complex number with its multiplicative inverse is 1. ${\displaystyle z{\frac {1}{z}}=1}$ Why bother showing these properties? We need to make sure that the complex numbers are "self-sufficient." That is, they satisfy most of the properties of real numbers we are all familiar with, with one additional caveat foreign to the real number system: ${\displaystyle i^{2}=-1,}$ which is what makes the complex numbers unique. The properties that have been laid out in the last two steps are needed to call the complex numbers a "field." For example, if there is no such thing as a multiplicative inverse of a complex number, then we cannot define what division is. Although a rigorous concept of a field is beyond the scope of this article, basically, the idea is that the properties shown above must be true in order for things in the complex plane to work out for all complex numbers, just like the field of real numbers. Luckily, these concepts are all intuitive in the reals, so they can easily be extended to the complex numbers. EditPolar Form Recall the coordinate transformations from Cartesian (rectangular) coordinates to polar coordinates. On the real coordinate plane, coordinates can either be rectangular or polar. In the Cartesian system, any point can be labeled with a horizontal and a vertical component. In the polar system, a point is labeled with the distance from the origin (the magnitude) and the angle from the polar axis. Such coordinate transformations are given below. ${\displaystyle x=r\cos \theta }$ ${\displaystyle y=r\sin \theta }$ Looking at the diagram above, the complex number ${\displaystyle z}$ has two pieces of information defining it: ${\displaystyle r}$ and ${\displaystyle \phi .}$ ${\displaystyle r}$ is called the modulus of the number, while ${\displaystyle \phi }$ is called the argument. Rewrite the complex number in polar form. Substituting, we have the expression below. ${\displaystyle z=r(\cos \theta +i\sin \theta )}$ This is the complex number in polar form. We have its magnitude ${\displaystyle r={\sqrt {x^{2}+y^{2}}}}$ on the outside. Inside the parentheses, we have the trigonometric components, related to the Cartesian coordinates by ${\displaystyle \theta =\tan ^{-1}{\frac {y}{x}}.}$ Sometimes, the expression inside the parentheses is written as ${\displaystyle \operatorname {cis} \theta ,}$ which is an abbreviation for "cosine plus i sine." Compact the notation by using Euler's formula. Euler's formula ${\displaystyle e^{i\theta }=\cos \theta +i\sin \theta }$ is one of the most useful relations in complex analysis because it fundamentally links exponentiation to trigonometry. The next part of this article gives a visualization of the complex exponential function, while the classic series derivation is given in the tips. ${\displaystyle z=re^{i\theta }}$ Right now, you may ask, how can any complex number be represented as some number times an exponential? The reason is that because complex exponentials are rotations in the complex plane, the ${\displaystyle e^{i\theta }}$ term gives us the information about the angle. Rewrite the complex conjugate in polar coordinates. We know that on the complex plane, the conjugate is simply a reflection over the real axis. That means that the ${\displaystyle \cos \theta }$ part is unchanged, but the ${\displaystyle \sin \theta }$ changes sign. ${\displaystyle {\bar {z}}=r(\cos \theta -i\sin \theta )}$ When we compact the notation using Euler's formula, we find that the sign of the exponent is negated. ${\displaystyle {\bar {z}}=re^{-i\theta }}$ Revisit multiplication and division using polar notation. Recall from part 2 that, while addition and subtraction in Cartesian coordinates were straightforward, the other arithmetic operations were quite clumsy. In polar coordinates, however, they are made much easier. To multiply two complex numbers is to multiply their moduli and add their arguments. We can do this because of the properties of exponents. ${\displaystyle z_{1}z_{2}=(r_{1}e^{i\theta _{1}})(r_{2}e^{i\theta _{2}})=r_{1}r_{2}e^{i(\theta _{1}+\theta _{2})}}$ To divide two complex numbers is to divide their moduli and subtract their arguments. ${\displaystyle {\frac {z_{1}}{z_{2}}}={\frac {r_{1}e^{i\theta _{1}}}{r_{2}e^{i\theta _{2}}}}={\frac {r_{1}}{r_{2}}}e^{i(\theta _{1}-\theta _{2})}}$ Geometrically speaking, this makes complex numbers a lot easier to grasp, and simplifies pretty much everything associated with complex numbers in general. EditVisualization of the Exponential Function Understand the color wheel plot of a complex function. Complex functions require four dimensions to fully visualize their behavior, because a complex number is made up of two real parts. However, we can skirt past this obstacle by using hue and brightness as our parameters. The brightness is the absolute value (modulus) of the output of the function. The plot of the exponential function below defines black to be 0. The hue is the angle (argument) of the output of the function. One convention is to define red as the angle ${\displaystyle \theta =0.}$ Then, in increments of ${\displaystyle {\frac {\pi }{3}},}$ the color goes from yellow, green, cyan, blue, magenta, to red again, across the color wheel. Visualize the exponential function. The complex plot of the exponential function gives insights into how it can possibly be related to the trigonometric functions. When we restrict ourselves to the real axis, the brightness goes from dark (near 0) in the negatives, to light in the positives, as expected. When we restrict ourselves to the imaginary axis, however, the brightness stays the same, but the hue changes periodically, with a period of ${\displaystyle 2\pi .}$ This means that the complex exponential ${\displaystyle e^{i\theta }}$ is periodic in the imaginary direction. This is to be expected from Euler's formula, because the trigonometric functions ${\displaystyle \cos \theta }$ and ${\displaystyle \sin \theta }$ are periodic with periods of ${\displaystyle 2\pi }$ each as well. EditTips In step 4 of part 3, we compacted the polar form using Euler's formula, but this formula at first glance looks nonintuitive. A derivation of Euler's formula is given below. Recall that the real-valued function ${\displaystyle e^{x}}$ can be written in terms of a Taylor series. ${\displaystyle e^{x}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\cdots }$ Taylor series for cosine and sine can similarly be given. ${\displaystyle \cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots }$ ${\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-\cdots }$ What does the exponential function mean in terms of complex numbers? We define it with the complex number ${\displaystyle z.}$ ${\displaystyle e^{z}=1+z+{\frac {z^{2}}{2!}}+{\frac {z^{3}}{3!}}+\cdots }$ Everything is good if ${\displaystyle z}$ is a purely real number. We simply recover the usual exponential function. But what if ${\displaystyle z=iy}$ was a purely imaginary number? We get the following simplifications, because we remember that ${\displaystyle i^{2}=-1.}$ ${\displaystyle {\begin{aligned}e^{iy}&=1+iy+{\frac {(iy)^{2}}{2!}}+{\frac {(iy)^{3}}{3!}}+{\frac {(iy)^{4}}{4!}}+\cdots \\&=\left(1-{\frac {y^{2}}{2!}}+{\frac {y^{4}}{4!}}+\cdots \right)+i\left(y-{\frac {y^{3}}{3!}}+{\frac {y^{5}}{5!}}-\cdots \right)\\&=\cos y+i\sin y\end{aligned}}}$

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