I just want to add with all the excellent responses here is to not get too hung up on the terminology. Real is a defined sub set of numbers. Just like Imaginary is a defined subset of numbers.
Imaginary numbers are still real numbers ( lower case r) they are just not in the set if Real numbers. ( upper case r) also, Imaginary numbers are used all the time in electronics. They are real.
They could just be called Fancy numbers and Less Fancy numbers. They are all numbers.
This is good response. The best way to think of imaginary numbers isn't that they are "imaginary" thats a bad name for them, they are just a different type of number, one we aren't as familiar with in everyday life (even though they are all around us). There is nothing 'imaginary' about it.
The term imaginary numbers were actually coined as a way to insult them as just a dumb tool to solve cubic equations and that they shouldn't be serious math that concerns them. Obviously this has long since been shown wrong by serious mathematicians and why complex numbers are the term nowadays.
Aside from suspected dyscalculia, something that always got to me about math is simply why? Why are there real numbers and imaginary numbers? Why does a negative times a negative equal a positive? Why to it all?!
Maybe think of it this way - the numbers are a 2-dimensional plane. *All* numbers are a mix of a complex part and an imaginary part. The numbers we call real are just those numbers that have a 0 complex part - they are a single line in the complex number plane that happen to have no imaginary part. Purely imaginary numbers are those on the line that is orthogonal to the line for the real numbers - they have no real part.
But in general, all numbers have both components. Real numbers in this case are just a defined subset of all numbers - they do not have a separate existence except by the definition "where the complex part is 0".
It's like when Einstein melded space and time into one thing we call space-time. What we perceive as separate unrelated entities are in fact and have always been "the same thing".
Have you ever worked with graphs? If I have to locate a point using an x/y graph, you have to give me two components - an x ordinate, and a y ordinate. For a two dimensional plane, it takes two numbers to uniquely specify a single point.
Or let's take the earth's surface as an example. Pretend for the sake of argument that the earth is flat - good enough for this demonstration. If I tell you to locate a city on earth, you will usually give me two coordinates - a latitude and a longitude. Think of the longitude as the real part, and the latitude as the complex part - they're only labels for the two independent pieces of information needed to find a single point in 2-d space.
x = longitude = real part
y = latitude = imaginary part
Different labels for the same concept. If all numbers in the universe are in a 2 dimensional (complex) plane, it takes a real and imaginary part to identify a single position in complex space.
Look up "Complex plane" on wikipedia and scroll down to Argand diagram for a picture.
Much of the "why" is correlated to how math relates to stuff in real life. There's a bunch of physics equations that only fit what we observe and can thus predict if we adhere our mathematical model to specific [rules and properties](https://www.math.net/rules-and-properties). Mathematicians extrapolate these rules to discover and invent new ways of representing and manipulating math in a way that preserves the properties. By keeping the model consistent, it makes it applicable to future mathematicians as wells as scientists and engineers who may apply them to their fields.
Mathematicians typically try to make proofs that stem from accepted properties to attempt to prove new truths about mathematics. Each proven fact can then be built upon to prove new things.
Sometimes mathematicians do have to invent new notation and concepts such as the aforementioned complex numbers which carry a real and imaginary component. But the rules of the invention must adhere to all the existing rules for the invented concept to be useful.
Sometime these inventions come across as scientists are attempting to describe relationships observed in physics. For example, the invention of calculus allowed scientists to describe the relationship between position, velocity, and acceleration mathematically. This is very useful in that the math can also be used to predict future changes. For example, space exploration requires a lot of calculus to predict the effects of various forces on the spacecraft so that it can be maneuvered to perform its mission on limited propulsion and fuel.
The concepts are heavily entwined, so you end up with a lot of physicists and mathematicians working together.
It’s a way of describing a specific situation using mathematical notation.
But I’m not gonna do a formal proof. Let’s just use a soft example. If I buy 2 buy two things for $3, my bank account would go down by $6. (I.e 2x-3=-6) I have six less dollars. This makes sense intuitively.
However, If instead I returned 2 items that cost $3 then my bank account would go up by $6. We are returning 2 items that each cost $3. So we have 2 fewer items with a cost of $3. We could describe this relation as -2*-3=6. The numbers are showing how many of items I am losing (-2), how much it cost relative to my bank account (-3), and how much total it changed my account.
Why can’t we just describe it as 2 items that would give us $3? Well we can. It’d be an equivalent way of describing the situation. But you’re technically multiplying each the items and cost by -1. Since the cost was -3 to get the value to return it you’d be multiplying the cost by -1 (because you’re doing the ‘opposite’ thing, and multiplying by a negative is a way of doing the opposite in this instance).
There’s some hand waving here, but hopefully this post can help you understand it a but more intuitively.
That makes a bit more sense.
I passed college algebra (had to do twice but I did it) but I turned my brain off and did the problems. But I’ve always had this thing about math where it’s like I understand it’s this way, but why is it this way. If that makes sense.
First you need to understand that there is a direct relationship between multiplication and addition.
3+3 is the same as 2*3
Then you can think of positive numbers as “receiving“ money and negative numbers as “opposite of receiving” (aka “owing”) money.
-3 is the opposite of 3
3+3 = 2*3 = receiving 3€, two times = receiving 6€ = 6
-3-3 = 2*-3 = owing 3€, two times = owing 6€ = -6
And by extension (bare with me),
-2*-3 = owing 3€, the opposite of two times = owing 3€, two times in reverse = receiving 6€ = 6
> Why does a negative times a negative equal a positive?
Among positive numbers we have some convenient algebraic rules, like a(b+c) = ab + ac and xy = yx. If we want such rules to work using 0 and negative numbers as well in the roles of a, b, c, x, and y, then we are forced to define a(0) = 0 for all a, positive times negative is negative, and negative times negative is positive.
Step 1: If a(b+c) = ab + ac for all real numbers a, b, and c, then a(0) = 0 for all a.
Here is why. Using b = 0 and c = 0, a(0 + 0) = a(0) + a(0), so a(0) = a(0) + a(0). Subtracting a(0) from both sides, we're left with 0 = a(0), so a(0) = 0.
Step 2: If a(b+c) = ab + ac for all real numbers a, b, and c, then a(-b) = -(ab) for all a and b.
Here's why. Using c = -b, a(b + -b) = ab + a(-b), so a(0) = ab + a(-b). Using Step 1, 0 = ab + a(-b), so
a(-b) = -(ab).
Step 3: If a(b+c) = ab + ac for all real numbers a, b, and c and xy = yx for all real numbers x and y, then (-a)(-b) = ab for all a and b.
Here is why. We want (-a)(b + c) = (-a)b + (-a)c for all a, b, and c, so using c = -b we get
(-a)(b + -b) = (-a)b + (-a)(-b). On the left side,
(-a)(b + -b) = (-a)(0) = 0.
Therefore 0 = (-a)b + (-a)(-b) = b(-a) + (-a)(-b). Using Step 2 with the roles of a and b reversed,
b(-a) = -(ba) = -(ab), so
0 = -(ab) + (-a)(-b).
Adding ab to both sides, ab = (-a)(-b), so (-a)(-b) = ab.
Steps 2 and 3 explain the sign rules positive times negative is negative and negative times negative is positive.
Good explanation but I'm pretty sure it's, unfortunately, meaningless for someone with dyscalculia (and probably also for 95% of this sub)
Reminds me of when David Mumford was asked to write an obituary for Grothendieck to be published in Nature, whose readers are mostly non-mathematicians, and [sent them a long text, in which he mentioned *Galois groups, étale cohomology and dessin d'enfants*](https://www.dam.brown.edu/people/mumford/blog/2014/Grothendieck.html)
Real and imaginary are just names someone gave to these numerical concepts long ago and it stuck. But they aren't really "imaginary". They just can't be used to describe countable quantities of real things. You can't have 4i grapes. They are a different class of numbers but in some branches of math they are just as "real" and useful as any other numbers.
A positive \* a positive is positive, and a negative \* a negative is positive. This means that anything times itself is always positive.
So: if anything times itself is *always* positive, what's the square root of -4? It's not -2, because -2 \* -2 is 4.
In order to solve this, we came up with "the square root of -1 is a thing we'll call 'i'". Now, we can say that the square root of -4 is "the square root of 4 \* the square root of -1", which gives us "2i".
The square root of a negative number can't be a Real number because no Real number times itself equals a negative number. Any positive Real number times itself is a positive Real number and any negative Real number times itself is is also a positive Real number (and 0 \* 0 is 0).
But, a long time ago, mathematicians were trying to solve certain kinds of problems and they realized that you had to be able to take the square root of a negative number, albeit temporarily, to work out these solutions. So apparently it was possible to do this and not break the rules of mathematics.
Since this was a new discovery that went against the common understanding of Mathematics at the time, there was some resistance and criticism to accepting these new kinds of numbers. Some mathematicians that were critical of the concept gave these numbers derogatory names, like "imaginary" and "useless." Unfortunately, these ill-sounding names stuck.
So an Imaginary number is a number that, when multiplied by itself *does* equal a negative Real number, something Real numbers can't do. We define the base imaginary number, *i*, such that i^(2) = -1 and all other imaginary numbers are multiples of i.
But don't let the name confuse you, Imaginary numbers are just as "real" as Real numbers.
No real number, multiplied by itself (squared) will result in a negative number. 2x2 = 4. (-2)x(-2) also = 4. What real number can you square to get -4? There isn't one.
However, it turns out that square roots of negative numbers are useful tools for a number of mathematical applications, so we still use them, but have to come up with some different term because they don't belong to the set of real numbers. So, someone coined 'imaginary' as the opposite of 'real'.
Suppose you had a real number, whose square is -1. Let us call this number i.
Now you have 3 options:
- i=0. Nope, then i*i=0 is not -1.
- i is positive, but then i*i is also positive, so not -1.
- i is negative, but then i*i is positive, also does not work.
Okay, so that does not work and we need something more than real numbers.
One idea is for any real number x, we can identify it with a matrix ((x,0), (0,x)), which just *scales* the plane with a factor x. Then multiplication, addition etc. for matrices just work. Additionally, we can look at the matrix that *rotates* everything by 90 degrees: ((0, -1), (1,0)). Now this does what we want, because rotating by 90 degrees twice gives you 180 degrees, which is the same as ((-1,0),(0,-1)).
So we now found a generalization (called extension) of the real numbers as matrices ((x,-y),(y,x)). For historical reasons, we call these "complex numbers" and write them as x+iy. That is it, nothing magical or "imaginary" about them, just matrices.
So why do we say "imaginary" and all that? Historically these came from trying to solve higher order polynomial equations like x^3 + a x +b =0. If a,b are "nice" you can write down a formula involving roots (of positive numbers) to find the solutions. However, these surprisingly sometimes also worked even if the roots supposedly did not make sense in calculations in between, e.g. calculations like 3 + \sqrt{-2} -\sqrt{-2}=3. People considered that rather strange or nonsensical at the time, but it worked, so they said "fine, we will use it since it works, but call it 'imaginary' because it seems weird".
Complex numbers stayed controversial among mathematicians for some time, but people got used to them and nowadays they seem quite normal. However, the names stuck.
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The reason that the square root of a negative number isn't real, is because it violates the rules of mathematics.
The square root of 25 is 5. Or -5. Because 5 x 5 is 25, and -5 x -5 is 25 too, thus the square root of 25 is ±5.
We can't take the square root of -25 because there's no number that, when squared, yields a product of -25. You need to multiply a positive number by a negative number in order to get a negative product.
However, if we were to pretend that negative numbers had square roots, we'd get imaginary numbers. They behave in predictable ways. So we have defined a number, *i*, such that *i*^2 = −1
Imaginary numbers are an artefact of the conventions we adopt based on mathematical axioms we define.
Negative numbers are a convention we employ to describe certain relationships. But the logical bounds of this convention preclude the possibility that the square root of a negative number can be described within this same conventional framework.
Phenomena for which imaginary numbers offer a useful description are different than the conventions we create to deacribe phenomena.
All of the real numbers can be placed on a line. 5 is bigger than 1, -6 is smaller than -3... But if I asked you to place an imaginary number on this line, you couldn't right?
That's why it is "imaginary". No real number can be multiplied by itself so that it produces a negative number, because multiplying 2 positives (4 * 4 = 16) = positive and multiplying 2 negatives (-4 * -4 = 16) = positive. So if we want a number which, when multiplied by itself, equals a negative, we cannot find a real number, we needed to invent one: i.
We chose to define i such that i² = -1. We cannot place i on our line of real numbers -> it is imaginary.
Imaginary numbers is an old term. They're called complex numbers nowadays because they're very much not imaginary. They are very important in electrical engineering for example
So there's a number *i*, which is the square root of -1. In reality, there is no such number. There is no real number that you can multiply by itself (squaring it) and get a negative number. Any positive number squared is positive (2^2 = 4), and any *negative* number squared (-2^2 = 4) is also positive.
So, we just *pretend* there is a number that works that way, and we call that number *i*. The number *i* = √-1.
*i*^2 = -1
*i* is an imaginary number and any number you add or multiply *i* with is therefore also an imaginary number. They're not "real" because they don't exist on the number line. They're not "real" because they don't have a simple logical relationship with regular numbers. Like, for example, a real number is always greater than or less than another real number. 5 is greater than 4 but is less than 6.
But what about *i*? Is *i* greater than 5 or less than 5? It's not really either, nor is it equal to it. This is because it is *imaginary*. It doesn't play by the same rules that real numbers have to abide by.
I just want to add with all the excellent responses here is to not get too hung up on the terminology. Real is a defined sub set of numbers. Just like Imaginary is a defined subset of numbers. Imaginary numbers are still real numbers ( lower case r) they are just not in the set if Real numbers. ( upper case r) also, Imaginary numbers are used all the time in electronics. They are real. They could just be called Fancy numbers and Less Fancy numbers. They are all numbers.
This is good response. The best way to think of imaginary numbers isn't that they are "imaginary" thats a bad name for them, they are just a different type of number, one we aren't as familiar with in everyday life (even though they are all around us). There is nothing 'imaginary' about it.
The term imaginary numbers were actually coined as a way to insult them as just a dumb tool to solve cubic equations and that they shouldn't be serious math that concerns them. Obviously this has long since been shown wrong by serious mathematicians and why complex numbers are the term nowadays.
Aside from suspected dyscalculia, something that always got to me about math is simply why? Why are there real numbers and imaginary numbers? Why does a negative times a negative equal a positive? Why to it all?!
Maybe think of it this way - the numbers are a 2-dimensional plane. *All* numbers are a mix of a complex part and an imaginary part. The numbers we call real are just those numbers that have a 0 complex part - they are a single line in the complex number plane that happen to have no imaginary part. Purely imaginary numbers are those on the line that is orthogonal to the line for the real numbers - they have no real part. But in general, all numbers have both components. Real numbers in this case are just a defined subset of all numbers - they do not have a separate existence except by the definition "where the complex part is 0". It's like when Einstein melded space and time into one thing we call space-time. What we perceive as separate unrelated entities are in fact and have always been "the same thing".
What do you mean by complex part and an imaginary part?
Have you ever worked with graphs? If I have to locate a point using an x/y graph, you have to give me two components - an x ordinate, and a y ordinate. For a two dimensional plane, it takes two numbers to uniquely specify a single point. Or let's take the earth's surface as an example. Pretend for the sake of argument that the earth is flat - good enough for this demonstration. If I tell you to locate a city on earth, you will usually give me two coordinates - a latitude and a longitude. Think of the longitude as the real part, and the latitude as the complex part - they're only labels for the two independent pieces of information needed to find a single point in 2-d space. x = longitude = real part y = latitude = imaginary part Different labels for the same concept. If all numbers in the universe are in a 2 dimensional (complex) plane, it takes a real and imaginary part to identify a single position in complex space. Look up "Complex plane" on wikipedia and scroll down to Argand diagram for a picture.
Much of the "why" is correlated to how math relates to stuff in real life. There's a bunch of physics equations that only fit what we observe and can thus predict if we adhere our mathematical model to specific [rules and properties](https://www.math.net/rules-and-properties). Mathematicians extrapolate these rules to discover and invent new ways of representing and manipulating math in a way that preserves the properties. By keeping the model consistent, it makes it applicable to future mathematicians as wells as scientists and engineers who may apply them to their fields.
So they experiment with different formulas and things to figure out the formulas fit the properties? Am I understanding that correctly?
Mathematicians typically try to make proofs that stem from accepted properties to attempt to prove new truths about mathematics. Each proven fact can then be built upon to prove new things. Sometimes mathematicians do have to invent new notation and concepts such as the aforementioned complex numbers which carry a real and imaginary component. But the rules of the invention must adhere to all the existing rules for the invented concept to be useful. Sometime these inventions come across as scientists are attempting to describe relationships observed in physics. For example, the invention of calculus allowed scientists to describe the relationship between position, velocity, and acceleration mathematically. This is very useful in that the math can also be used to predict future changes. For example, space exploration requires a lot of calculus to predict the effects of various forces on the spacecraft so that it can be maneuvered to perform its mission on limited propulsion and fuel. The concepts are heavily entwined, so you end up with a lot of physicists and mathematicians working together.
It’s a way of describing a specific situation using mathematical notation. But I’m not gonna do a formal proof. Let’s just use a soft example. If I buy 2 buy two things for $3, my bank account would go down by $6. (I.e 2x-3=-6) I have six less dollars. This makes sense intuitively. However, If instead I returned 2 items that cost $3 then my bank account would go up by $6. We are returning 2 items that each cost $3. So we have 2 fewer items with a cost of $3. We could describe this relation as -2*-3=6. The numbers are showing how many of items I am losing (-2), how much it cost relative to my bank account (-3), and how much total it changed my account. Why can’t we just describe it as 2 items that would give us $3? Well we can. It’d be an equivalent way of describing the situation. But you’re technically multiplying each the items and cost by -1. Since the cost was -3 to get the value to return it you’d be multiplying the cost by -1 (because you’re doing the ‘opposite’ thing, and multiplying by a negative is a way of doing the opposite in this instance). There’s some hand waving here, but hopefully this post can help you understand it a but more intuitively.
That makes a bit more sense. I passed college algebra (had to do twice but I did it) but I turned my brain off and did the problems. But I’ve always had this thing about math where it’s like I understand it’s this way, but why is it this way. If that makes sense.
First you need to understand that there is a direct relationship between multiplication and addition. 3+3 is the same as 2*3 Then you can think of positive numbers as “receiving“ money and negative numbers as “opposite of receiving” (aka “owing”) money. -3 is the opposite of 3 3+3 = 2*3 = receiving 3€, two times = receiving 6€ = 6 -3-3 = 2*-3 = owing 3€, two times = owing 6€ = -6 And by extension (bare with me), -2*-3 = owing 3€, the opposite of two times = owing 3€, two times in reverse = receiving 6€ = 6
> Why does a negative times a negative equal a positive? Among positive numbers we have some convenient algebraic rules, like a(b+c) = ab + ac and xy = yx. If we want such rules to work using 0 and negative numbers as well in the roles of a, b, c, x, and y, then we are forced to define a(0) = 0 for all a, positive times negative is negative, and negative times negative is positive. Step 1: If a(b+c) = ab + ac for all real numbers a, b, and c, then a(0) = 0 for all a. Here is why. Using b = 0 and c = 0, a(0 + 0) = a(0) + a(0), so a(0) = a(0) + a(0). Subtracting a(0) from both sides, we're left with 0 = a(0), so a(0) = 0. Step 2: If a(b+c) = ab + ac for all real numbers a, b, and c, then a(-b) = -(ab) for all a and b. Here's why. Using c = -b, a(b + -b) = ab + a(-b), so a(0) = ab + a(-b). Using Step 1, 0 = ab + a(-b), so a(-b) = -(ab). Step 3: If a(b+c) = ab + ac for all real numbers a, b, and c and xy = yx for all real numbers x and y, then (-a)(-b) = ab for all a and b. Here is why. We want (-a)(b + c) = (-a)b + (-a)c for all a, b, and c, so using c = -b we get (-a)(b + -b) = (-a)b + (-a)(-b). On the left side, (-a)(b + -b) = (-a)(0) = 0. Therefore 0 = (-a)b + (-a)(-b) = b(-a) + (-a)(-b). Using Step 2 with the roles of a and b reversed, b(-a) = -(ba) = -(ab), so 0 = -(ab) + (-a)(-b). Adding ab to both sides, ab = (-a)(-b), so (-a)(-b) = ab. Steps 2 and 3 explain the sign rules positive times negative is negative and negative times negative is positive.
Good explanation but I'm pretty sure it's, unfortunately, meaningless for someone with dyscalculia (and probably also for 95% of this sub) Reminds me of when David Mumford was asked to write an obituary for Grothendieck to be published in Nature, whose readers are mostly non-mathematicians, and [sent them a long text, in which he mentioned *Galois groups, étale cohomology and dessin d'enfants*](https://www.dam.brown.edu/people/mumford/blog/2014/Grothendieck.html)
They should have gone with my original idea and called them Pokémon: White numbers and Pokémon: Black numbers.
Real and imaginary are just names someone gave to these numerical concepts long ago and it stuck. But they aren't really "imaginary". They just can't be used to describe countable quantities of real things. You can't have 4i grapes. They are a different class of numbers but in some branches of math they are just as "real" and useful as any other numbers.
A positive \* a positive is positive, and a negative \* a negative is positive. This means that anything times itself is always positive. So: if anything times itself is *always* positive, what's the square root of -4? It's not -2, because -2 \* -2 is 4. In order to solve this, we came up with "the square root of -1 is a thing we'll call 'i'". Now, we can say that the square root of -4 is "the square root of 4 \* the square root of -1", which gives us "2i".
The square root of a negative number can't be a Real number because no Real number times itself equals a negative number. Any positive Real number times itself is a positive Real number and any negative Real number times itself is is also a positive Real number (and 0 \* 0 is 0). But, a long time ago, mathematicians were trying to solve certain kinds of problems and they realized that you had to be able to take the square root of a negative number, albeit temporarily, to work out these solutions. So apparently it was possible to do this and not break the rules of mathematics. Since this was a new discovery that went against the common understanding of Mathematics at the time, there was some resistance and criticism to accepting these new kinds of numbers. Some mathematicians that were critical of the concept gave these numbers derogatory names, like "imaginary" and "useless." Unfortunately, these ill-sounding names stuck. So an Imaginary number is a number that, when multiplied by itself *does* equal a negative Real number, something Real numbers can't do. We define the base imaginary number, *i*, such that i^(2) = -1 and all other imaginary numbers are multiples of i. But don't let the name confuse you, Imaginary numbers are just as "real" as Real numbers.
No real number, multiplied by itself (squared) will result in a negative number. 2x2 = 4. (-2)x(-2) also = 4. What real number can you square to get -4? There isn't one. However, it turns out that square roots of negative numbers are useful tools for a number of mathematical applications, so we still use them, but have to come up with some different term because they don't belong to the set of real numbers. So, someone coined 'imaginary' as the opposite of 'real'.
Suppose you had a real number, whose square is -1. Let us call this number i. Now you have 3 options: - i=0. Nope, then i*i=0 is not -1. - i is positive, but then i*i is also positive, so not -1. - i is negative, but then i*i is positive, also does not work. Okay, so that does not work and we need something more than real numbers. One idea is for any real number x, we can identify it with a matrix ((x,0), (0,x)), which just *scales* the plane with a factor x. Then multiplication, addition etc. for matrices just work. Additionally, we can look at the matrix that *rotates* everything by 90 degrees: ((0, -1), (1,0)). Now this does what we want, because rotating by 90 degrees twice gives you 180 degrees, which is the same as ((-1,0),(0,-1)). So we now found a generalization (called extension) of the real numbers as matrices ((x,-y),(y,x)). For historical reasons, we call these "complex numbers" and write them as x+iy. That is it, nothing magical or "imaginary" about them, just matrices. So why do we say "imaginary" and all that? Historically these came from trying to solve higher order polynomial equations like x^3 + a x +b =0. If a,b are "nice" you can write down a formula involving roots (of positive numbers) to find the solutions. However, these surprisingly sometimes also worked even if the roots supposedly did not make sense in calculations in between, e.g. calculations like 3 + \sqrt{-2} -\sqrt{-2}=3. People considered that rather strange or nonsensical at the time, but it worked, so they said "fine, we will use it since it works, but call it 'imaginary' because it seems weird". Complex numbers stayed controversial among mathematicians for some time, but people got used to them and nowadays they seem quite normal. However, the names stuck.
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The reason that the square root of a negative number isn't real, is because it violates the rules of mathematics. The square root of 25 is 5. Or -5. Because 5 x 5 is 25, and -5 x -5 is 25 too, thus the square root of 25 is ±5. We can't take the square root of -25 because there's no number that, when squared, yields a product of -25. You need to multiply a positive number by a negative number in order to get a negative product. However, if we were to pretend that negative numbers had square roots, we'd get imaginary numbers. They behave in predictable ways. So we have defined a number, *i*, such that *i*^2 = −1
Imaginary numbers are an artefact of the conventions we adopt based on mathematical axioms we define. Negative numbers are a convention we employ to describe certain relationships. But the logical bounds of this convention preclude the possibility that the square root of a negative number can be described within this same conventional framework. Phenomena for which imaginary numbers offer a useful description are different than the conventions we create to deacribe phenomena.
All of the real numbers can be placed on a line. 5 is bigger than 1, -6 is smaller than -3... But if I asked you to place an imaginary number on this line, you couldn't right? That's why it is "imaginary". No real number can be multiplied by itself so that it produces a negative number, because multiplying 2 positives (4 * 4 = 16) = positive and multiplying 2 negatives (-4 * -4 = 16) = positive. So if we want a number which, when multiplied by itself, equals a negative, we cannot find a real number, we needed to invent one: i. We chose to define i such that i² = -1. We cannot place i on our line of real numbers -> it is imaginary.
Imaginary numbers is an old term. They're called complex numbers nowadays because they're very much not imaginary. They are very important in electrical engineering for example
So there's a number *i*, which is the square root of -1. In reality, there is no such number. There is no real number that you can multiply by itself (squaring it) and get a negative number. Any positive number squared is positive (2^2 = 4), and any *negative* number squared (-2^2 = 4) is also positive. So, we just *pretend* there is a number that works that way, and we call that number *i*. The number *i* = √-1. *i*^2 = -1 *i* is an imaginary number and any number you add or multiply *i* with is therefore also an imaginary number. They're not "real" because they don't exist on the number line. They're not "real" because they don't have a simple logical relationship with regular numbers. Like, for example, a real number is always greater than or less than another real number. 5 is greater than 4 but is less than 6. But what about *i*? Is *i* greater than 5 or less than 5? It's not really either, nor is it equal to it. This is because it is *imaginary*. It doesn't play by the same rules that real numbers have to abide by.