BuBu 08 Dec 2019, 04:47
Is it true, that all numbers can be made, as the sum of complex numbers, but only the ones with the argument of 45^"o" 135^"o" and 270^"o"?
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Is it true, that all numbers can be made, as the sum of complex numbers, but only the ones with the argument of 45^"o" 135^"o" and 270^"o"?
Relevant page
<a href="https://www.tankonyvtar.hu/en/tartalom/tamop412A/2011-0038_25_juhasz_diszkret_matematika/ch02s03.html">
Diszkrét matematika |
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What I've done so far
.Murray 09 Dec 2019, 01:22
@BuBu: You haven't indicated any working so that I can get a sense of where you are having trouble.
To start, are you able to form the 11 integers `-5,-4,...4,5` as the sum of complex numbers, but only the ones with the argument of 45^"o" 135^"o" and 270^"o"?
X
@BuBu: You haven't indicated any working so that I can get a sense of where you are having trouble. To start, are you able to form the 11 integers `-5,-4,...4,5` as the sum of complex numbers, but only the ones with the argument of 45^"o" 135^"o" and 270^"o"?
stephenB 23 Dec 2019, 01:49
I'll try.
The `45^"o"` ones will be like `1+i`.
The `135^"o"` ones will be like `-1+j`.
The `270^"o"` ones will be like `-j`
So we can form:
`-5 = 5*((-1+j) + (-j))`
`-4 =4*((-1+j) + (-j))`
`-3 = 3*((-1+j) + (-j))`
`-2 = 2*((-1+j) + (-j))`
`-1 = (-1+j) + (-j)`
`0 = (-1+j)+(1+j)+(-j)+(-j)`
`1 = (1+j) + (-j)`
`2 = 2*((1+j) + (-j))`
`3 = 3*((1+j) + (-j))`
`4 = 4*((1+j) + (-j))`
`5 = 5*((1+j) + (-j))`
So it works for the integers `-5,-4,-3,...5.`
We could do similar things with the decimals, so I'm inclined to think this would be possible for all real numbers.
X
I'll try. The `45^"o"` ones will be like `1+i`. The `135^"o"` ones will be like `-1+j`. The `270^"o"` ones will be like `-j` So we can form: `-5 = 5*((-1+j) + (-j))` `-4 =4*((-1+j) + (-j))` `-3 = 3*((-1+j) + (-j))` `-2 = 2*((-1+j) + (-j))` `-1 = (-1+j) + (-j)` `0 = (-1+j)+(1+j)+(-j)+(-j)` `1 = (1+j) + (-j)` `2 = 2*((1+j) + (-j))` `3 = 3*((1+j) + (-j))` `4 = 4*((1+j) + (-j))` `5 = 5*((1+j) + (-j))` So it works for the integers `-5,-4,-3,...5.` We could do similar things with the decimals, so I'm inclined to think this would be possible for all real numbers.
Murray 23 Dec 2019, 01:56
Looks good to me. So yes, as long as we are talking about the reals, I think the claim is reasonable.
X
Looks good to me. So yes, as long as we are talking about the reals, I think the claim is reasonable.
justairplane 28 Feb 2024, 04:44
I encountered this problem again. It's good that it was solved here.
X
I encountered this problem again. It's good that it was solved here.
usoperating 20 Nov 2025, 06:16
Re: google baseball
From what I understand, if you restrict yourself to complex numbers with arguments only at 45°, 135°, and 270°, you won’t be able to reach all complex numbers as sums. You can generate numbers along specific directions, but the full complex plane requires being able to combine numbers in more varied directions. You might be able to cover certain lattices or regions, but not every possible complex number. It could help to try plotting a few sums and see which areas of the plane are reachable.
X
Re: <a href="https://googlebaseball.io/">google baseball</a> From what I understand, if you restrict yourself to complex numbers with arguments only at 45°, 135°, and 270°, you won’t be able to reach all complex numbers as sums. You can generate numbers along specific directions, but the full complex plane requires being able to combine numbers in more varied directions. You might be able to cover certain lattices or regions, but not every possible complex number. It could help to try plotting a few sums and see which areas of the plane are reachable.
X
nice
arrayballoon 24 Nov 2025, 21:56
re: snow rider
Yes, that’s exactly right! When dividing complex numbers, we multiply both the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part (the “complex” component) from the denominator. This process leaves a real number in the denominator, which makes the expression simpler and easier to work with — similar to how we remove radicals in regular fractions.
X
re: <a href="https://snow-rider3d.io/">snow rider</a> Yes, that’s exactly right! When dividing complex numbers, we multiply both the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part (the “complex” component) from the denominator. This process leaves a real number in the denominator, which makes the expression simpler and easier to work with — similar to how we remove radicals in regular fractions.
DereecePhillips 26 Nov 2025, 23:24
@BuBu, it seems like you're exploring the idea of representing integers as the sum of complex numbers with specific arguments. Murray's point about showing your working is crucial for pinpointing where the challenge lies. Have you tried breaking down how to express those 11 integers using complex numbers with angles of 45°, 135°, and 270°? It might be a fascinating exercise to work through examples to see if patterns emerge based on these arguments. Exploring different approaches step by step could shed light on this intriguing concept further.
X
@BuBu, it seems like you're exploring the idea of representing integers as the sum of complex numbers with specific arguments. Murray's point about showing your working is crucial for pinpointing where the challenge lies. Have you tried breaking down how to express those 11 integers using complex numbers with angles of 45°, 135°, and 270°? It might be a fascinating exercise to work through examples to see if patterns emerge based on these arguments. Exploring different approaches step by step could shed light on this intriguing concept further.
DereecePhillips 26 Nov 2025, 23:24
@BuBu, it seems like you're exploring the idea of representing integers as the sum of complex numbers with specific arguments. Murray's point about showing your working is crucial for pinpointing where the challenge lies. Have you tried breaking down how to express those 11 integers using complex numbers with angles of 45°, 135°, and 270°? It might be a fascinating exercise to work through examples to see if patterns emerge based on these arguments. Exploring different approaches step by step could shed light on this intriguing concept further.
X
@BuBu, it seems like you're exploring the idea of representing integers as the sum of complex numbers with specific arguments. Murray's point about showing your working is crucial for pinpointing where the challenge lies. Have you tried breaking down how to express those 11 integers using complex numbers with angles of 45°, 135°, and 270°? It might be a fascinating exercise to work through examples to see if patterns emerge based on these arguments. Exploring different approaches step by step could shed light on this intriguing concept further.
PaulDickerson 28 Nov 2025, 22:02
Re: Drift Boss
If magnitudes are restricted (e.g. only integer multiples, or you may only add unit vectors). Then you do not reach every complex number. For example, if you may only sum unit vectors (no scaling), reachable points lie on a discrete lattice of sums of those unit vectors; many targets are not achievable. Similarly, integer-only coefficients produce a discrete cone / lattice, not the whole plane.
X
Re: <a href="https://driftbossgame.io">Drift Boss</a> If magnitudes are restricted (e.g. only integer multiples, or you may only add unit vectors). Then you do not reach every complex number. For example, if you may only sum unit vectors (no scaling), reachable points lie on a discrete lattice of sums of those unit vectors; many targets are not achievable. Similarly, integer-only coefficients produce a discrete cone / lattice, not the whole plane.
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Re: Slope Game
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Re: <a href="https://slopegame-online.com">Slope Game</a> During my undergrad days, I struggled to understand how different forms of regression could give seemingly contradictory results. It felt like I was perpetually on a slippery Slope of understanding, always sliding back a bit. Share your similar slope experiences!
Conway 10 Dec 2025, 23:21
Great post! Complex numbers can be so interesting, especially when looking at their sums and how they can combine in unique ways. I love how math can sometimes feel like a puzzle to solve. Speaking of puzzles, if you're a fan of games that really get you thinking, you might enjoy “Fnaf.” It combines strategy and suspense in a thrilling way. Check it out Fnaf for some fun! Keep up the great discussions!
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