Imaginary Numbers Are Real [Part 1: Introduction]

Imaginary Numbers Are Real [Part 1: Introduction] Let’s say we’re given the function f(x)=x^2 + 1. We can graph our function and get a nice parabola. Now let’s say we want to figure out where the equation equals zero we want to find the roots. On our plot this should be where the function crosses the x-axis. As we can see, our parabola actually never crosses the x-axis, so according to our plot, there are no solutions to the equation x^2+1=0. But there’s a small problem. A little over 200 years ago a smart guy named Gauss proved that every polynomial equation of degree n has exactly n roots. Our polynomial has a highest power, or degree, of two, so we should have two roots. And Gauss’ discovery is not just some random rule, today we call it the FUNDAMENTAL THEOREM OF ALGEBRA. So our plot seems to disagree with something so important it’s called the FUNDAMENTAL THEOREM OF ALGEBRA, which might be a problem. What Guass is telling us here, is that there are two perfectly good values of x that we could plug into our function, and get zero out. Where could these 2 missing roots be? The short answer here is that we don’t have enough numbers. We typically think of numbers existing on a 1 dimensional continuum – the number line. All our friends are here: 0, 1, negative numbers, fractions, even irrational numbers like root 2 show up. But this system is incomplete. And our missing numbers are not just further left or right, they live in a whole new dimension. Algebraically, this new dimension has everything to do with a problem that was mathematically considered impossible for over two thousand years: the square root of negative one. When we include this missing dimension in our analysis, our parabola gets way more interesting. Now that our input numbers are in their full two dimensional form, we see how our function x^2+1 really behaves. And we can now see that our function does have exactly two roots! We were just looking in the wrong dimension. So, why is this extra dimension that numbers possess not common knowledge? Part of this reason is that it has been given a terrible, terrible name. A name that suggest that these numbers aren’t ever real! In fact, Gauss himself had something to say about this naming convention. So yes, this missing dimension is comprised of numbers that have been given ridiculous name imaginary. Gauss proposed these numbers should instead be given the name lateral so from here on, let’s let lateral mean imaginary. To get a better handle on imaginary, I mean, lateral numbers, and really understand what’s going on here, let’s spend a little time thinking about numbers. Early humans really only had use for the natural numbers, that is 1, 2, 3, and so on. This makes sense because of how numbers were used. So to early humans, the number line would have just been a series of dots. As civilizations advanced, people needed answers to more sophisticated math questions – like when to plant seeds, how to divide land, and how to keep track of financial transactions. The natural numbers just weren’t cutting it anymore, so the Egyptians innovated and developed a new, high tech solution: fractions. Fractions filled in the gaps in our number line, and were basically cutting edge technology for a couple thousand years. The next big innovations to hit the number line were the number zero and negative numbers, but it took some time to get everyone on board. Since it’s not obvious what these numbers mean or how they fit into the real world, zero and negative numbers were met with skepticism, and largely avoided or ignored. Some cultures were more suspicious than others, depending largely on how people viewed the connection between mathematics and reality. And this is not all ancient history – just a few centuries ago, mathematicians would intentionally move terms around to avoid having negatives show up in equations. Suspicion of zero and negative numbers did eventually fade – partially because negatives are useful for expressing concepts like debt, but mostly because negatives just kept sneaking into mathematics. It turns out there’s just a whole lot of math you just can’t do if you don’t allow negative numbers to play. Without negatives, simple algebra problems like x + 3=2 have no answer. Before negatives were accepted, this problem would have no solution, just like we thought our original problem had no solution. The thing is, it’s not crazy or weird to think problems like this have no solutions – in words, this algebra problem basically says: “if I have 2 things and I take away 3, how many things do I have left?” It’s not surprising that most of the people who have lived on our planet would be suspicious of questions like this. These problems don’t make much sense. Even brilliant mathematicians of the 18th century, such as Leonard Euler, didn’t really know what to do with negatives he at one point wrote that negatives were greater than infinity. So it’s fair to say that negative and imaginary numbers raise a lot of very good, very valid questions. Like why do we require students to understand and work with numbers that eluded the greatest mathmatical minds for thousands of years? Why did we even come accept negative and imaginary numbers in the first place, when they don’t really seem connected to anything in the real world? And how do these extra numbers help explain the missing solutions to our problem? Next time, we’ll begin to address these questions by going way back to the discovery of imaginary numbers.

100 thoughts on “Imaginary Numbers Are Real [Part 1: Introduction]

  1. -1+0^0/2+1. Qed. RH.TM. PROVEN!!!
    Eureka I Found It! $$$.
    New Viral System Cranks Out Leads and Cash Daily!

    Go Trump 2020 MAGA. KAG. Note: Money is NOT the root of all Evil! It is the LOVE of money, that is the root of all evil, or GREED! Earn money, then take a part of your wealth & help others in need to get successful too. Don't forget to feed the poor, etc… Shalom. Mathew Chapter, 25.

  2. I know this doesn't have anything to do with the video, but I would like to share these questions and this message with as many people as possible.

    1. What do you think happens to someone after they die?

    2. Do you think there is an afterlife?

    3. If heaven exists, how do you get there?

    4. Would you be good enough to get into heaven?

    Here are a few questions to see if you are a good person:

    1. How many lies have you told in your whole life?

    2. Have you ever stolen?

    3. Have you ever used God's name in vain?

    4. Jesus said that "whoever looks at a woman to lust for her has already committed adultery with her in his heart" (Matthew 5:28). Have you ever looked with lust?

    5. If you have broken any of these commandments, would you be innocent or guilty on Judgment Day?

    6. Would you go to Heaven or Hell?

    The Bible says that "all liars shall have their part in the lake which burns with fire and brimstone" (Revelation 21:8) and "Those who indulge in sexual sin, or who worship idols, or commit adultery, or are male prostitutes, or practice homosexuality, or are thieves, or greedy people, or drunkards, or are abusive, or cheat people — none of these will inherit the Kingdom of God" (1 Corinthians 6:9-10). So, if you are guilty of any of the ten commandments, then you are in trouble when Judgment Day comes! Do you know what God did for guilty sinners so that we wouldn't have to go to Hell?

    2,000 years ago, God became a human being, Jesus of Nazareth, who suffered and died on the cross for our sins and rose from the dead on the third day, defeating death. The punishment that we deserve for breaking God's law, Jesus took in our place. He paid the fine that we could never pay so that God can legally set us free by forgiving our sins. He is just, by punishing evil through Jesus’ death on the cross, while being rich in mercy towards us.

    What you have to do is repent, or turn from your sin, and trust alone in Jesus. Don't trust your own goodness, because, as Jesus said, “No one is good — except God alone” (Mark 10:18). Trust alone in the savior. The moment you do that, God will forgive every sin you have ever committed and grant you eternal life, and He will give you a new heart with new desires, so that you will love to do what is right. You must repent and trust in Jesus.

    When are you going to do that?

    Here is a conversation between a Christian and an atheist on this subject:

  3. I will prove that -x > infinity:

    x + x = x
    Let x be minus infinity,
    -inf – inf = -inf
    So x is -inf.
    Let x be infinity.
    Inf + inf =inf
    So x is infinity.
    Therefore minus infinity=infinity.
    Negatives are more than minus infinity.
    So -x>infinity.

  4. this guy sees imaginary things in real life .. someone give this guy a prescription pill from the local drugstore please.

  5. From now on when people tell me to thank positively I will say " there can be no solutions without negatives being part of the equation."

  6. Mathematics are ontological. Real. Materialism limits even our view of mathematics. Imaginary numbers might be our link to the soul. We’re living equations that are both local and non local. Zero, infinity and imaginary numbers represent the soul, mind and consciousness.

  7. Isso parece ser o mais lógico. Porém, está errado, porque ele não pode ser representado em 2/3 d. Por isso a matemática é chata. Estes valores tem que ser gradativamente, mostrados em um diferencial/planos e pontos de diferentes observadores para cer certo. E mesmo assim não explica com certeza os verdadeiros valores, pois os números são vivos e ao mesmo tempo não existem.

  8. I want to say that my underatanding of imaginary numbers was completely revolutionized within the first 30 secs of this video. I had no idea that "imaginary" numbers even had anything to do with roots or the Cartesian coordinate system. I had no idea why the hell 'i' had it's own thing. Thank you

  9. Wym everyone knows the square root of -1 is simplified to i u will never ever need to know anythjng more even if u are a rocket scientist

  10. Imaginary Numbers are BS. A complete ABSURDITY. The same goes for the Sign rule. No wonder Maths svck upto a point when you start mixing apples with bananas. Period.

  11. 3:40 WRONG..! Olmeca culture did not erected pyramids whatsoever, besides that Is Chichen Itza pyramid an It Is mayan, not Olmeca.

  12. Negatives being greater than infinity actually comes up in computer science.
    If you store an integer type number, then -1 will be just 1s. Every bit is 1. -2 would be every bit as one, but the last is a zero. And so on.
    The reason for this is so you can add and subtract numbers the exact way you'd do with positive numbers.
    -1 (every bit is a one) + 1 (only last bit is one) = 0 (every bit is zero, because it all rolls over).

  13. Your history is a little lopsided to shit on white people… of course. Where was the only negative criticism? Exactly.

  14. I watched this near the end of 5th grade and had no idea what f(x) or a parabola was. Now I’m in high school and I know everything in this video.

  15. Yeah, so leave my imaginary friends alone. Even if…no one ever sees them. Including me. And they never call. Or text. Or anything.

    Starts to cry

  16. I can't imagine how negative numbers weren't used before. I'd imagine people used it but in a different way. Such as, instead of saying negative one they just said you owe me one or I'm short one. Something like that but idk.

  17. Well, apart from the fact that you didn't even make a label for the Perso-Islamic scientific culture (which is basically responsible for the birth of algebra as we know it), it is a good video.

  18. This is quantum insanity at its finest. Pulling extra dimensions out of your ass is not math and it’s not real. Math describes, if it can’t describe something, you don’t make up dimensions and imaginary numbers. You call it a phenomenon and try to understand in a way that doesn’t involve counting.

  19. I see negative as a convention. All measurements need a point of reference. In Real numbers it's 0. Everything under is deemed negative. It has correspondence in real life in temperature degrees, bank accounts, speed variation, and so on.Our life is full of minuses.

  20. Why does in the 3D graph, the parabola goes DOWN than y=+1? (blue areas)??
    It ONLY crosses y=+1 on a 2D graph but why does it keep going down on a 3D graph???

  21. Well about negatives, you could say that indeed I can't take you more than what you have, so I give you a loan.
    What could you say about imaginary numbers?
    Where can you put an example for it?
    Also, like negatives, is it true that solutions that usually use imaginary numbers can be achieved without using them?

  22. I feel like many things will get easier if the naming system is more "normal" friendly.
    but this is much like the simple speak (?) from the Big Brother in that book (forgot the name)

  23. What is this lateral/imaginary plane at 1:49 called? Also how did Gauss come up with conclusion that any polynomial of a N degree should have N roots?

  24. It is not Indian and Persian but only Indian. Indians have a huge history with maths and many other complicated subjects to their greatest depths

  25. It can be correct that negatives are greater than infinity as euler said.simce infinity exists in 2 type.there are infinite numbers but also there can be infinite numbers between 1 and 2 if go on with the decimal.

  26. I have a little question, If I consider imaginary no. in the z- direction. How can I plot the 3d graph of y=x^+1 like yours…

  27. Стоп, значит Артур Шарифов простой перевёл и рассказал на русском?

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