Should I aspire for rationality or am I too stupid?
...
...consistently do relatively well in Physics, Chemistry, Engineering and Programming modules.
You are not too stupid.
I’m in a double degree in Chemical Engineering and Business and on track to receive First Class Honours in both.
You are really, really, seriously, not too stupid.
Yet, I find it difficult to multiply two 2-digit numbers in my head.
That’s something that you might want to work on, but it’s not a general intelligence failure. There are some tricks that can be learned (or discovered) and employed to multiply by specific numbers more quickly; alternatively, practice will help to speed up your mental multiplication.
I just really don’t get why I don’t do well in math, which I assume would be the best measure of one’s fluid intelligence.
Scholastic math is a different beast. I can say that a lot of professors have issues with the “standard” math curriculum. I have taught university calculus myself and I don’t think that the curriculum and textbook I had to work with had much to do with “fluid intelligence”.
It seems that my mind lights up with too many questions when I learn math, many of which are difficult to answer. (My professor does not have much time to meet students for consultations and I don’t think I want to waste his time). It seems that I need to undergo suspension of disbelief just to do math, which doesn’t seem right given that a lot of it has been rigorously proven by loads of people much smarter than me.
Sounds like one source for your troubles. It’s a lot harder to succeed at school math and go through the motions if you have unanswered questions about why the method works (and aren’t willing to blindly follow formulas). By all means bring your questions up to the professor. If he’s teaching, there’s probably some university policy that he be available to students for a certain amount of hours outside of class (i.e. it’s part of his job). You lose nothing by trying. Even an e-mail wouldn’t be a bad idea in the last resort. In my experience, professors tend to complain about students who never seek help until they show up the day before the final at their wits’ end (or, worse still, after the final to ask why they failed). By that point it’s too late.
Things such as why dividing by zero doesn’t work confuses me
We like our multiplication rules to work nicely and division by zero causes problems. There’s no consistent way to define something like 0⁄0 (you could say that since 1 x 0 = 0, 0⁄0 should be 1, but this argument works for any number). With something like 1⁄0, you could say “infinity”, but does that then mean 0 x infinity = 1? What’s 2⁄0 then?
A very easy way to improve your writing would be to separate your text into paragraphs.
It doesn’t take any intelligence but just awareness of norms.
It seems that my mind lights up with too many questions when I learn math, many of which are difficult to answer. (My professor does not have much time to meet students for consultations and I don’t think I want to waste his time).
Not everybody is good at math. That’s okay. Scott Alexander who’s an influential person in this community writes on his blog:
In Math, I just barely by the skin of my teeth scraped together a pass in Calculus with a C-. [...]“Scott Alexander, who by making a herculean effort managed to pass Calculus I, even though they kept throwing random things after the little curly S sign and pretending it made sense.”[...]I don’t want to have to accept the blame for being a lazy person who just didn’t try hard enough in Math.
Things such as why dividing by zero doesn’t work confuses me and I often wonder at things such as the Fundamental Theorem of Calculus.
Math is about abstract thinking. That means “common sense” often doesn’t work. One has to let go of naive assumptions and accept answers that don’t seem obvious.
In many cases the ability to trust that established mathematical finding are correct even if you can’t follow the proof that establishes them is an useful ability. It makes life easier.
In many cases the ability to trust that established mathematical finding are correct even if you can’t follow the proof that establishes them is an useful ability. It makes life easier.
While yes, that can make life easier, it also means that if the reason why you can’t follow the proof is because you’re misunderstanding the finding in question, then you’re not applying any error checking and anything that you do that depends on that misunderstanding is going to potentially be incorrect. So, if you’re going into any field where mathematics is important, it can also make life significantly harder.
It’s hard to put in words what I mean. There a certain ability to think in abstract concepts that you need in math. Wanting things to feel like you “understand” can be the wrong mode of engaging complex math.
That doesn’t mean that understanding math isn’t useful but it’s abstract understanding and trying to seek a feeling of common sense can hold people back.
I… think I learnt math in a very different way to you. If I didn’t feel that I understood something, I went back until I felt that I did.
I do not understand the difference between an “abstract understanding” and a “feeling of common sense”. Is a feeling of common sense not a subtype of an abstract understanding (in the same way that a “square” is a subtype of a “rectangle”)?
On the contrary, failing to feel common sense is usually a sign that you don’t really understand what’s going on. Your understanding of an abstract concept is only as good as that of your best example. The abstract method in mathematics is just a way of taking features common to several examples and formulating a theory that can be applied in many cases. With that said, it is a useful skill in math to be able to play the game and proceed formally.
There’s an anecdote about a famous math professor who had to teach a class. The first time, the students didn’t understand. A year later, he taught it again. Learning from experience, he made it simpler. The students still didn’t understand. When he taught it a third time, he made it simple enough that even he finally understood it.
I will concede that in practice it can be expedient to trust the experts with the complications and use ready-made formulas.
I noticed that many articles in the sequences confuse me at times because I can think of multiple interpretations of a particular paragraph but have no idea which was intended. Also, many actions/thoughts of Harry in HPMOR confuse me. I might have interpretations of the events but I don’t think those interpretations are likely to be correct. Is this normal?
This seems normal to me. What is intended is very often not an easy question to answer.
I have edited the post though, I think that saying that I am on track to receive First Class Honours in both is too optimistic.
The mere fact that you have been accepted for and expect to pass a double degree tells me that you are really not too stupid. (I’m not actually sure what the difference between Second Upper and First Class Honours is—I assume that’s because you’re referring to the education system of a country with which I am not familiar).
I just really don’t get why I don’t do well in math, which I assume would be the best measure of one’s fluid intelligence. Things such as why dividing by zero doesn’t work confuses me and I often wonder at things such as the Fundamental Theorem of Calculus. It seems that my mind lights up with too many questions when I learn math, many of which are difficult to answer.
Theory: You had a poor teacher in primary-school level maths, and failed to learn something integral to the subject way back there. Something really basic and fundamental. Despite this severe handicap, you have managed to get to the point where you’re going to pass a double degree (which implies good things about your intelligence).
Is this normal too?
I… don’t actually know. Throughout my entire school career, I was the guy for whom maths came easily. I don’t know what’s normal there.
Actually, it may be possible to narrow down what you’re missing in mathematics. (If we do find it, it won’t solve all your math problems immediately, but it’ll be a good first step)
Let’s start here:
Things such as why dividing by zero doesn’t work confuses me
My summary (intended as an incentive to read the Feynman, not a replacement for
reading it):
We start with addition of discrete objects (“I have two apples; you have
three apples. How many apples do we have between us?”). No fractions, no
negative numbers, no problem.
We get other operations by repetition—multiplication is repeated
addition, exponentiation is repeated multiplication.
We get yet more operations by reversal—subtraction is reversed
addition, division is reversed multiplication, roots and logarithms are
reversed exponentiation. These operations also let us define new kinds of
numbers (fractions, negative numbers, reals, complex numbers) that are not
necessarily useful for counting apples or sheep or pebbles but are useful
in other contexts.
Rules for how to work with these new kinds of numbers are motivated by
keeping things as consistent as possible with already-existing rules.
Well, About 3-5 percent of the best students in a cohort can expect to get First Class Honours. It basically means 97th percentile, or 95th percentile, depending on the quality of the students. The 75th to 95th percentile can expect to get Second Class Honours.
Which implies that I can, tentatively, estimate you to be in the top 10% of people who are accepted for a degree. That’s really good.
I must admit that this question stunned me. I don’t actually know.
...I think we’ve found the start of the problem. Your foundations have a few holes.
Dividing X by Y, at its core, means that I have X objects, I want to place them in Y exactly equal piles, how many objects do I place per pile? (At least, that’s the definition I’d use). In this way, the usefulness of the operation is immediately apparent; if I have six apples, and I want to divide them among three people, I can give each person two apples.
I can use the same definition if I have five apples and three people; then I give each person one and two-thirds apples.
This also works for negative numbers; if I have negative-six apples (i.e. a debt of six apples) I can divide that into three piles by placing negative-two apples in each pile.
Division by zero then becomes a matter of taking (say) six apples, and trying to put them into zero piles. (I hope that makes the problem with division by zero clear).
And yes, there is a fancy algorithm that I can put X and Y in and get the quotient out… but that algorithm is not a particularly good basic definition of division. (Interestingly, I note that your definition jumps straight to setting out separate cases and then trying to apply a different algorithm to each individual case. This would make it very hard to work with in practice; I’ve worked with division algorithms on computers, and they’re far simpler, conceptually, than what you had there. If that’s what you’ve been working with, then I am really not surprised that you’ve been having trouble with maths).
Now let’s see how far this goes...
Define “multiplication”, “addition”, and “subtraction”.
When I read your answer, I was thinking, (seriously no offense because I know you are really smart) I don’t know for sure that this definition works for complex numbers.
It does; complex numbers are just another type of number. We’ll get to them shortly.
And then I was thinking that mathematics relies on definitions and deductive reasoning and intuition cannot give the certainty of deductive reasoning, thus it might be a fallacy to think that something simple and intuitive is an accurate model of mathematical reality… then I remembered that it was taught in kindergartens even...
To be fair, sometimes the intuitive answer is wrong; one does have to take care. But sometimes, as in these cases, the intuitive model does work.
Define “multiplication”
X*Y : I have Y sets of X objects, how many objects do I have?
Exactly.
“addition”
X+Y : I have X objects. I am given Y objects. How many objects do I have?
Perfect.
It’s easy to visualize imaginary numbers as another type of object ‘x’, and I am given y objects. So I have x + y imaginary objects and X + Y real objects.
You could do it that way, and it leads to the correct answers, but I think it’s fundamentally problematic to see complex numbers as intrinsically different to real numbers. (For one thing, real numbers are a subset of complex numbers in any case).
“subtraction”
X-Y : I have X objects. Y objects are taken away from me.
Right.
Then it makes me wonder what other exceptions to manipulation there is
There’s only one that I can think of off the top of my head; if x^z=y^z, this does not mean that x=y (i.e. we can’t just take the z’th root on both sides of the equation). This can be clearly demonstrated with x=2, y=-2 and z=2. Two squared is four, which is equal to (negative two) squared, but two is not equal to negative two.
Now, as to complex numbers. Let me start by asking you to define a “complex number”.
Okay, those are all—well, I think I can kind of see some relation to complex numbers in there, but it’s very vague.
So, let me describe how I understand complex numbers. To do that, we’ll have to go right back to the very basics of mathematics; numbers.
Imagine, for a moment, an infinite piece of paper. (Or you can get a piece of paper and draw this, if you like; you won’t need to draw the whole, infinite thing, just enough to get the idea)
Take a point, nice and central. Mark it “zero”.
Select a second point (traditionally, this point is chosen to the right of zero, but the location doesn’t matter). Mark it “one”.
Now, let us call the distance between zero and one a “jump”. You start from zero, you move a jump in a particular direction, you get to “one”. You move another jump in the same direction, you get to “two”. Another jump, “three”. Another jump, “four”. And so on, to infinity. These are the positive integers.
Now, consider an operation; addition. If I apply addition to any pair of positive integers, I get another positive integer. Any of these numbers that I add gives me a number I already have; I can add no new numbers with addition.
However, I can also invert the addition operation, to get subtraction. If I want to find X+Y, I hop X jumps from the zero point,then Y more jumps. But if I want to find X-Y, I must jump X jumps to the right, then Y jumps to the left; and this gives me the negative integers. Add them to the mental numberline.
At this point, multiplication gives us no new numbers. Division , however, does.
You will now notice, there are still gaps between the numbers. To fill these gaps, we turn to division; X/Y gives us a plethora of new numbers (1/2, 2⁄3, 3⁄4, 4⁄5, so on and so forth), hundreds and millions and billions of little dots between each point on the numberline. These are the rational numbers.
Is the numberline full yet? Hardly; it turns out that the rational numbers are so small a proportion of the numberline that it’s still more empty space than marked point. I could say that there’s billions of irrational numbers for every rational number, but that severely underestimates the number of irrational numbers that there are.
But let’s add all the irrational numbers as well. (If you’re actually drawing this, just take a ruler and draw a line across the page, such that all your integers fall on the line).
This line, then, is the famous numberline. I’m sure you’ve seen it before, on classroom walls and similar. It contains all the real numbers and, now that we’ve added the irrational numbers, it is full; there is no space on the line where another number can be added.
Now, let’s consider squaring. The square of one is one. The square of any positive number greater than one is an even greater positive number (for example, two squared is four). The square of any positive number between zero and one is a positive number closer to zero (0.5 squared is 0.25).
The square of zero is zero.
The square of any negative number is equal to the square of the corresponding positive number; thus the square of negative two is four.
Therefore, four has two square roots; 2 and −2. Similarly, one’s square roots are one and minus one.
So, a question then emerges; where are the square roots of minus one?
They cannot be on the numberline. There is no space for new numbers on the line, and the square of every number on the line is a positive number (or zero).
Let us call the square roots of minus one i and -i (somwhat arbitrary notation that was used once and stuck) Where do we put them on the line?
Since the line is full, we cannot put them on the line. If you place the line such that the zero is in front of you, the positive numbers head off to the right, and the negative numbers go to the left, then i is found one jump directly up from zero. Similarly, -i is one jump directly down from zero.
So, they are numbers, but they are not on the number line.
And now that we have placed i and -i, we can apply the same operation as we used earlier.
Addition: adding 1 is a jump to the right. Similarly, adding i is a jump upwards. There is a 2i two jumps above zero; a 3i three jumps above zero, and so on.
In fact, by following the same steps as were used to construct the original, real number line, we can create an imaginary number line at right angles to it; so that we can point to, say, 2.5i, or even pi i.
Then, if we want t find the point where (say) 3+4i is, we first jump three jumps to the right, then we jump four jumps up; adding the numbers 3 and 4i. 3+4i is thus a clearly defined point on the numberplane (since it’s no longer one-dimensional, “numberline” is not exactly accurate anymore).
Adding and subtracting complex numbers on this plane is perfectly straightforward (though actually describing what i apples look like is beyond me). Multiplication follows the rules for multiplying additive expressions; that is, (a+b)*(c+d) = ac+ad+bc+bd. So, therefore:
...
You are not too stupid.
You are really, really, seriously, not too stupid.
That’s something that you might want to work on, but it’s not a general intelligence failure. There are some tricks that can be learned (or discovered) and employed to multiply by specific numbers more quickly; alternatively, practice will help to speed up your mental multiplication.
Scholastic math is a different beast. I can say that a lot of professors have issues with the “standard” math curriculum. I have taught university calculus myself and I don’t think that the curriculum and textbook I had to work with had much to do with “fluid intelligence”.
Sounds like one source for your troubles. It’s a lot harder to succeed at school math and go through the motions if you have unanswered questions about why the method works (and aren’t willing to blindly follow formulas). By all means bring your questions up to the professor. If he’s teaching, there’s probably some university policy that he be available to students for a certain amount of hours outside of class (i.e. it’s part of his job). You lose nothing by trying. Even an e-mail wouldn’t be a bad idea in the last resort. In my experience, professors tend to complain about students who never seek help until they show up the day before the final at their wits’ end (or, worse still, after the final to ask why they failed). By that point it’s too late.
We like our multiplication rules to work nicely and division by zero causes problems. There’s no consistent way to define something like 0⁄0 (you could say that since 1 x 0 = 0, 0⁄0 should be 1, but this argument works for any number). With something like 1⁄0, you could say “infinity”, but does that then mean 0 x infinity = 1? What’s 2⁄0 then?
A very easy way to improve your writing would be to separate your text into paragraphs. It doesn’t take any intelligence but just awareness of norms.
Math.stackexchange exists for that purpose.
Not everybody is good at math. That’s okay. Scott Alexander who’s an influential person in this community writes on his blog:
Math is about abstract thinking. That means “common sense” often doesn’t work. One has to let go of naive assumptions and accept answers that don’t seem obvious.
In many cases the ability to trust that established mathematical finding are correct even if you can’t follow the proof that establishes them is an useful ability. It makes life easier.
In addition to what CCC wrote http://math.stackexchange.com/questions/26445/division-by-0 is a good explanation of the case.
Accepting feedback and directly applying it is great :)
While yes, that can make life easier, it also means that if the reason why you can’t follow the proof is because you’re misunderstanding the finding in question, then you’re not applying any error checking and anything that you do that depends on that misunderstanding is going to potentially be incorrect. So, if you’re going into any field where mathematics is important, it can also make life significantly harder.
It’s hard to put in words what I mean. There a certain ability to think in abstract concepts that you need in math. Wanting things to feel like you “understand” can be the wrong mode of engaging complex math.
That doesn’t mean that understanding math isn’t useful but it’s abstract understanding and trying to seek a feeling of common sense can hold people back.
I… think I learnt math in a very different way to you. If I didn’t feel that I understood something, I went back until I felt that I did.
I do not understand the difference between an “abstract understanding” and a “feeling of common sense”. Is a feeling of common sense not a subtype of an abstract understanding (in the same way that a “square” is a subtype of a “rectangle”)?
On the contrary, failing to feel common sense is usually a sign that you don’t really understand what’s going on. Your understanding of an abstract concept is only as good as that of your best example. The abstract method in mathematics is just a way of taking features common to several examples and formulating a theory that can be applied in many cases. With that said, it is a useful skill in math to be able to play the game and proceed formally.
There’s an anecdote about a famous math professor who had to teach a class. The first time, the students didn’t understand. A year later, he taught it again. Learning from experience, he made it simpler. The students still didn’t understand. When he taught it a third time, he made it simple enough that even he finally understood it.
I will concede that in practice it can be expedient to trust the experts with the complications and use ready-made formulas.
This doesn’t seem to be true for anything that’s normally analyzed statistically: the stock market, for example, or large-scale meteorology.
This seems normal to me. What is intended is very often not an easy question to answer.
The mere fact that you have been accepted for and expect to pass a double degree tells me that you are really not too stupid. (I’m not actually sure what the difference between Second Upper and First Class Honours is—I assume that’s because you’re referring to the education system of a country with which I am not familiar).
Theory: You had a poor teacher in primary-school level maths, and failed to learn something integral to the subject way back there. Something really basic and fundamental. Despite this severe handicap, you have managed to get to the point where you’re going to pass a double degree (which implies good things about your intelligence).
I… don’t actually know. Throughout my entire school career, I was the guy for whom maths came easily. I don’t know what’s normal there.
Actually, it may be possible to narrow down what you’re missing in mathematics. (If we do find it, it won’t solve all your math problems immediately, but it’ll be a good first step)
Let’s start here:
Define “division”.
I recommend chapter 22 (“Algebra”) of volume 1 of The Feynman Lectures on Physics. Here’s a PDF.
My summary (intended as an incentive to read the Feynman, not a replacement for reading it):
We start with addition of discrete objects (“I have two apples; you have three apples. How many apples do we have between us?”). No fractions, no negative numbers, no problem.
We get other operations by repetition—multiplication is repeated addition, exponentiation is repeated multiplication.
We get yet more operations by reversal—subtraction is reversed addition, division is reversed multiplication, roots and logarithms are reversed exponentiation. These operations also let us define new kinds of numbers (fractions, negative numbers, reals, complex numbers) that are not necessarily useful for counting apples or sheep or pebbles but are useful in other contexts.
Rules for how to work with these new kinds of numbers are motivated by keeping things as consistent as possible with already-existing rules.
Which implies that I can, tentatively, estimate you to be in the top 10% of people who are accepted for a degree. That’s really good.
...I think we’ve found the start of the problem. Your foundations have a few holes.
Dividing X by Y, at its core, means that I have X objects, I want to place them in Y exactly equal piles, how many objects do I place per pile? (At least, that’s the definition I’d use). In this way, the usefulness of the operation is immediately apparent; if I have six apples, and I want to divide them among three people, I can give each person two apples.
I can use the same definition if I have five apples and three people; then I give each person one and two-thirds apples.
This also works for negative numbers; if I have negative-six apples (i.e. a debt of six apples) I can divide that into three piles by placing negative-two apples in each pile.
Division by zero then becomes a matter of taking (say) six apples, and trying to put them into zero piles. (I hope that makes the problem with division by zero clear).
And yes, there is a fancy algorithm that I can put X and Y in and get the quotient out… but that algorithm is not a particularly good basic definition of division. (Interestingly, I note that your definition jumps straight to setting out separate cases and then trying to apply a different algorithm to each individual case. This would make it very hard to work with in practice; I’ve worked with division algorithms on computers, and they’re far simpler, conceptually, than what you had there. If that’s what you’ve been working with, then I am really not surprised that you’ve been having trouble with maths).
Now let’s see how far this goes...
Define “multiplication”, “addition”, and “subtraction”.
It does; complex numbers are just another type of number. We’ll get to them shortly.
To be fair, sometimes the intuitive answer is wrong; one does have to take care. But sometimes, as in these cases, the intuitive model does work.
Exactly.
Perfect.
You could do it that way, and it leads to the correct answers, but I think it’s fundamentally problematic to see complex numbers as intrinsically different to real numbers. (For one thing, real numbers are a subset of complex numbers in any case).
Right.
There’s only one that I can think of off the top of my head; if x^z=y^z, this does not mean that x=y (i.e. we can’t just take the z’th root on both sides of the equation). This can be clearly demonstrated with x=2, y=-2 and z=2. Two squared is four, which is equal to (negative two) squared, but two is not equal to negative two.
Now, as to complex numbers. Let me start by asking you to define a “complex number”.
Okay, those are all—well, I think I can kind of see some relation to complex numbers in there, but it’s very vague.
So, let me describe how I understand complex numbers. To do that, we’ll have to go right back to the very basics of mathematics; numbers.
Imagine, for a moment, an infinite piece of paper. (Or you can get a piece of paper and draw this, if you like; you won’t need to draw the whole, infinite thing, just enough to get the idea)
Take a point, nice and central. Mark it “zero”.
Select a second point (traditionally, this point is chosen to the right of zero, but the location doesn’t matter). Mark it “one”.
Now, let us call the distance between zero and one a “jump”. You start from zero, you move a jump in a particular direction, you get to “one”. You move another jump in the same direction, you get to “two”. Another jump, “three”. Another jump, “four”. And so on, to infinity. These are the positive integers.
Now, consider an operation; addition. If I apply addition to any pair of positive integers, I get another positive integer. Any of these numbers that I add gives me a number I already have; I can add no new numbers with addition.
However, I can also invert the addition operation, to get subtraction. If I want to find X+Y, I hop X jumps from the zero point,then Y more jumps. But if I want to find X-Y, I must jump X jumps to the right, then Y jumps to the left; and this gives me the negative integers. Add them to the mental numberline.
At this point, multiplication gives us no new numbers. Division , however, does.
You will now notice, there are still gaps between the numbers. To fill these gaps, we turn to division; X/Y gives us a plethora of new numbers (1/2, 2⁄3, 3⁄4, 4⁄5, so on and so forth), hundreds and millions and billions of little dots between each point on the numberline. These are the rational numbers.
Is the numberline full yet? Hardly; it turns out that the rational numbers are so small a proportion of the numberline that it’s still more empty space than marked point. I could say that there’s billions of irrational numbers for every rational number, but that severely underestimates the number of irrational numbers that there are.
But let’s add all the irrational numbers as well. (If you’re actually drawing this, just take a ruler and draw a line across the page, such that all your integers fall on the line).
This line, then, is the famous numberline. I’m sure you’ve seen it before, on classroom walls and similar. It contains all the real numbers and, now that we’ve added the irrational numbers, it is full; there is no space on the line where another number can be added.
Now, let’s consider squaring. The square of one is one. The square of any positive number greater than one is an even greater positive number (for example, two squared is four). The square of any positive number between zero and one is a positive number closer to zero (0.5 squared is 0.25).
The square of zero is zero.
The square of any negative number is equal to the square of the corresponding positive number; thus the square of negative two is four.
Therefore, four has two square roots; 2 and −2. Similarly, one’s square roots are one and minus one.
So, a question then emerges; where are the square roots of minus one?
They cannot be on the numberline. There is no space for new numbers on the line, and the square of every number on the line is a positive number (or zero).
Let us call the square roots of minus one i and -i (somwhat arbitrary notation that was used once and stuck) Where do we put them on the line?
Since the line is full, we cannot put them on the line. If you place the line such that the zero is in front of you, the positive numbers head off to the right, and the negative numbers go to the left, then i is found one jump directly up from zero. Similarly, -i is one jump directly down from zero.
So, they are numbers, but they are not on the number line.
And now that we have placed i and -i, we can apply the same operation as we used earlier.
Addition: adding 1 is a jump to the right. Similarly, adding i is a jump upwards. There is a 2i two jumps above zero; a 3i three jumps above zero, and so on.
In fact, by following the same steps as were used to construct the original, real number line, we can create an imaginary number line at right angles to it; so that we can point to, say, 2.5i, or even pi i.
Then, if we want t find the point where (say) 3+4i is, we first jump three jumps to the right, then we jump four jumps up; adding the numbers 3 and 4i. 3+4i is thus a clearly defined point on the numberplane (since it’s no longer one-dimensional, “numberline” is not exactly accurate anymore).
Adding and subtracting complex numbers on this plane is perfectly straightforward (though actually describing what i apples look like is beyond me). Multiplication follows the rules for multiplying additive expressions; that is, (a+b)*(c+d) = ac+ad+bc+bd. So, therefore:
(3+4i)*(2+5i) = (3*2)+(3*5i) + (4i*2) + (4i*5i) = 6 + 15i + 8i + 20*i*i
But since i is defined such that i*i=-1, that means:
(3+4i)*(2+5i) = 6 + 15i + 8i + 20(-1) = 23i-14
Voila, multiplication.