I keep googling it and I still don’t understand what they are and their differences 😭😭
In: 18
A root of an equation is a number where that equation is equal to zero. Some expressions, like x^2 – 4, have real roots. In this case the expression is zero when x = 2 or x = -2.
Other expressions, like x^2 + 4 have no real numbers that can make them equal zero. There are, however, complex numbers that can. Recall a complex number is a number that includes the imaginary unit i, where i = sqrt(-1). We can set x = 2i, and that means x^2 = -4. So the expression x^2 + 4 becomes -4 + 4 = 0 and we say that the expression has a complex root at x = 2i.
‘Real’ is a math definition. It basically means the regular normal numbers you know
‘Imaginary’ is a math definition. It sort of expands the Real numbers.
‘complex’ is a math definition. It’s the combination of Real and Imaginary. It’s sort of 2-dimensional numbers. One axis being the real and the other axis being the Imaginary. Look up some pictures of a complex plane.
A root of a number X is finding the number that you have to multiply with itself to find X. The most common is the square root, where you have to multiply a number with itself once to find the target X. The square root of 4 is 2 (and -2) because 2×2=4. (You can of course have ‘higher’ roots.) The difference arises when you try to get a root of a target number X that is negative. Let’s say you want the square root of -4. Is it 2? No because 2×2 is not -4. Is it -2? No because -2x-2=4.
Damn.
The solution to this are non-real numbers, namely imaginary numbers, which is a very difficult topic I know nothing about and is out of the scope of ELI5.
This means that the square root of -4 is non real, i.e. a non-real root.
To understand that you first need to understand what a real number is.
A natural number is all the numbers you can count. (1,2,3…).
A whole number does include the negative numbers and zero.
A rational number is any number that can be written as the ratio of two whole numbers. For example 1/3 or 22/7.
A real number is any number on the number line. Famous examples are pi and e, but also numbers like root(2) are real numbers but not rational numbers.
When taking the root you are asking “What number can I multiply with itself to get the number under the root?”.
If you are multiplying a positive number with itself you will get a positive result. If you multiply a negative number with itself it will also be a positive number (negative*negative=positive).
The answer to root(-1) seems impossible to find. And when you are looking at the real numbers it is. So mathematicians invented complex numbers. They said that root(-1)=i. You are no longer dealing with the number line but with a number plane.
When you take the root of any negative number you can get an answer on the number plane. There is a complex number (also known as an imaginary number, that might be where i is from).
Complex numbers are very useful in things like electronics, but in some fields you don’t want to look for complex answers, so you just say that this root doesn’t have an answer in the real numbers and you leave it at that.
If you ever see a big electric motor and you look at it’s plate you will see cos(phi) on there. That phi is the angle how far the apparent power is off the number line in complex math.
Real roots have no imaginary component, non real roots can have imaginary components
An example. The 4th roots of 1. You can multiply 1•1•1•1 to get 1. You can also multiply -1•-1•-1•-1 to get 1. Those are real roots.
Then there’s also i•i•i•i = 1 and -i•-i•-i•-i = 1 which are non real roots
An easy way to solve for all of them is polar form. Say you want to find the cube roots of 1. Well the polar form is 1∠0 which is equal to 1∠360 which is equal to 1∠720 (you can keep going, but you’ll just get repeated answers in further steps since we’re using the cube root which will have 3 roots)
Take those, and since it’s the cube root which is the third power, divide all of those angle by 3. You’ll end up with 1∠0 ; 1∠120 ; and 1∠240. If the angle is 0 or 180 it is a real root, if it is 90 or 270 it is a purely imaginary root, and anything else is a complex root. The way you get the final answer is converting the polar coordinates back to rectangular. So 1∠0 becomes 1•(cos(0) + sin(0) • i) = 1 {real root}; 1∠120 becomes 1•(cos(120) + sin(120) • i) = -0.5 + 0.866i; And 1∠240 becomes 1•(cos(240) + sin(240) • i) = -0.5 – 0.866i {both complex roots}
The reason it would repeat if you kept going is because if you go to 1∠1080 and divide 1080 by 3, you get 360 and 1∠0 would be equal to 1∠360 and just give you 1 again as a final answer
An easy way to understand these concepts is to look at the equation plotted on a graph. The real roots of an equation are the values where the line/curve intersects the x axis.
https://www.wolframalpha.com/input?i=x%5E2-3x+%2B1
You can see visually that the equation above crosses the x axis twice, so it has 2 real roots.
Now there is an interesting property of polynomials, which states that any polynomial of degree n (with n being the highest power), has n roots. The caveats being that:
– Roots can be repeated (double root, triple root, etc)
– Roots can be complex numbers
If you graph the equation and it doesn’t intersect the x axis, then it’s roots are complex.
If the curve at any point touches the x axis but doesn’t pass through, then it has a double root (or more generally a root repeated an even number of times)
To put it simply, we can create equations where there are no answers that work using our normal numbers.
To start, what do I mean by “normal numbers”? Well the math term for it is ‘real numbers’. These are numbers that you could use in every day life, and they themselves have a few different types (which I won’t go into here in detail), but these could include natural numbers like 1, 2 and 3, integers like -10, 0 and 100, rational numbers like 1/2 or 1/3 and irrational numbers like pi.
So the real numbers encompass a lot of different types of numbers, so surely they must be usable for all our equations, right? Unfortunately no, there are some equations where no real numbers will work. A basic example is the equation x^2 = -1
This equation is saying “what number can I multiply by itself to get -1?”. If you think back to the rules of positive and negative numbers, you’ll see where the issue is. If we multiply a positive number by a positive number, we get a positive number. If we multiply a negative number by a negative number, we still get a positive number. With what we know so far, the only way we can multiply two numbers together to get a negative answer is if one of the numbers is negative. But if one of the two numbers is positive and one is negative, then we can’t use that for our equation as the numbers need to be the same.
How do we solve this equation then? Based on the rules we know, it doesn’t seem like there is any real numbers that could work? And you would be right in thinking that. So mathematicians did what they often like to do, and they make something up to help fix this problem. Their solution was ‘imaginary numbers’, and these are numbers that are separate from the real numbers, so they behave differently. At their most basic level, we have an imaginary number that acts as an alternative to 1, except this number gives a negative answer when multiplied by itself. Specifically, this is a number called i, and is defined as i^2 = -1. This is the number that answers our equation from earlier, and this answer is what we call a non-real root of the equation x^2 = -1.
Although these imaginary numbers were created as sort of a band-aid to an issue we found in math, it turns out that they are incredibly useful in higher level math and engineering applications. There are many more complicated equations out there that may have a combination of real and non-real roots, and using these imaginary numbers helps solve a lot of other issues in mathematics and allows us to approach complex issues from other angles.
As a side note, one of the most profound instances of imaginary numbers is in something called Euler’s Identity, which is a formula that states if you raise the math constant e to the power of i times pi, and then add 1, you get 0. Or in math language: e^(i*pi) + 1 = 0 This simple formula is a bizarre example of how seemingly unrelated things in math can be linked together.
A quadratic equation is y = ax² + bx + c. How do you find a value for x, so that y will be 0? There’s a thing called the quadratic formula that does this for you. You take the numbers for a, b, and c, and put them in to this formula, and you’ll get the x you need to make y equal 0.
However, this quadratic formula has a square root. Sometimes the numbers you put into the formula mean that square root is for a negative number.
You can get the square root of say 25 which is 5. But what’s the square root of -25?
That’s where complex numbers step in. They let you say (i × 5) is the square root of -25. What is i? i is made up by mathematicians, to be the square root of -1. If you multiply i by itself – that is, square it – you’ll get -1. Or if you multiply (i × 5) by (i × 5), you’ll get (-1 × 25) = -25.
This lets you avoid shrugging and giving up when faced with negative square roots.
A real root is any time your quadratic formula has a positive square root. A non-real (aka complex) root is when your quadratic formula has a negative square root.
Edit: for accuracy’s sake, -5 is also a root of 25. And so, (i × -5) is also a root of -25. Handwaving to avoid adding extra complexity to this explanation, this is how you get two roots for the quadratic equation.
YouTube channel Welch Labs has [an incredible series of short, ELI5-level videos](https://www.youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF) on exactly this topic. I recommend you watch it.
Here’s some quick vocabulary:
– Expression: A calculation you can do if you know the values of all your variables.
– Polynomial: A certain kind of expression that’s a sum of terms, where each term is a constant times a variable raised to a whole-number power. Example: 2x^3 + 3x^2 – 14x – 15.
– Root: A number that makes an expression equal zero. Example: x = 2.5 makes the above expression 0, it turns into 31.25 + 18.75 – 35 – 15 which is 0.
– Real number: Numbers that are on the number line.
Most of the numbers you encounter in your life are real numbers. Examples of real numbers:
– The counting numbers (1, 2, 3, …) are real numbers
– Zero is a real number
– Fractions (1/3, 7/9, 1043/221) are real numbers
– Irrational numbers like π, e or √2 that can’t be expressed as fractions
– Negative numbers (-1, -2, -11/15)
There’s another number line perpendicular to the real number line. The numbers on that second number line are called “imaginary numbers.” The imaginary numbers are based on multiplying real numbers by √-1 or i for short.
So if you have an equation like x^2 = 4, you have two real solutions (x = 2 and x = -2). These correspond to the two real roots of the expression x^2 – 4. (To find this expression, you subtract one side of the equation to get something = 0, then the roots — the things that make the expression = 0 — are solutions to the equation.)
If you have an equation like x^2 = -4, there are no real solutions (you can’t square a real number and get a negative number). However there there are two imaginary solutions (x = 2i and x = -2i). These values (2i and -2i) are non-real roots of the expression x^2 + 4, just like 2 and -2 are real roots of x^2 – 4.
By adding real numbers and imaginary numbers you can form an entire plane called the complex plane. So the non-real roots might be complex, for example 2+3i and 2-3i are the roots of x^2 – 4x + 13.
So x³=8 what is x? You might have a few questions like can x be a set of numbers or just one? If we think about this question like we need a function that undoes the cubing or squaring there can be more answers. Like x²=4 both -2 and 2 squared is 4 so this function would have to output both -2 and 2 an an answer. That isn’t a function so we have to define what a root is. The root is always the positive.
If I ask it like this x²=4 what is x? x is -2 or 2. But if I ask sqrt(4)=x, x by definition is 2.
So now we have an idea and a solid definition about roots. The root is now a function with one parameter the kind of root you want squareroot, cuberoot…
Rational numbers are numbers you can write as a ratio of two intigers. ⅘, ⅔, ⅞… or 1, 2, 3 or -77. These are rational. Irrational numbers are numbers that cannot be expressed with a ratio between two intigers. The root is a process or now we can call it a function that sometimes outputs Irrational numbers even when you give it rational imputs. Like 2, sqrt(2) = well thats the thing. Thats the most compact form you can write it.
Roots are very misbehaved if you allow only whole numbers the outputs lead into real numbers and imaginary numbers.
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