A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can be represented by symbols, called numerals; for example, "5" is a numeral that represents the number five. As only a relatively small number of symbols can be memorized, basic numerals are commonly organized in a numeral system, which is an organized way to represent any number. The most common numeral system is the Hindu–Arabic numeral system, which allows for the representation of any number using a combination of ten fundamental numeric symbols, called digits. In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, a numeral is not clearly distinguished from the number that it represents.
In mathematics, the notion of a number has been extended over the centuries to include 0, negative numbers, rational numbers such as one half
(
1
2
)
{\displaystyle \left({\tfrac {1}{2}}\right)}
, real numbers such as the square root of 2
(
2
)
{\displaystyle \left({\sqrt {2}}\right)}
and π, and complex numbers which extend the real numbers with a square root of −1 (and its combinations with real numbers by adding or subtracting its multiples). Calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. Their study or usage is called arithmetic, a term which may also refer to number theory, the study of the properties of numbers.
Besides their practical uses, numbers have cultural significance throughout the world. For example, in Western society, the number 13 is often regarded as unlucky, and "a million" may signify "a lot" rather than an exact quantity. Though it is now regarded as pseudoscience, belief in a mystical significance of numbers, known as numerology, permeated ancient and medieval thought. Numerology heavily influenced the development of Greek mathematics, stimulating the investigation of many problems in number theory which are still of interest today.During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers, and may be seen as extending the concept. Among the first were the hypercomplex numbers, which consist of various extensions or modifications of the complex number system. In modern mathematics, number systems (sets) are considered important special examples of more general categories such as rings and fields, and the application of the term "number" is a matter of convention, without fundamental significance.
As I was thinking about understanding the nature of multiplication and division, understanding the nature of numbers, I thought about the idea that there is something beyond multiplication and repeated addition. Then I thought about dimensional multiplication operations, such as area and volume...
Any set of a series of numbers consisting of increasing integer members, all of which are determined by a common proposition or characteristic, will always be infinite in size.
Examples…
Prime numbers
Mersenne primes
Odd perfect numbers(if they exist)
Zeroes of the Zeta function
Regardless...
I'm trying to find an explicit example showing exactly how the U(1) “circle group” of complex numbers double-covers 2D planar rotations R(θ) that form the rotation group SO(2).
There are various explanations available online, some of which are clear but seem to be at variance with other...
I’m looking for someone to explain the first two equations in this article. I’ve got a good grip on it, but missing any sort of live feedback.
https://www.cs.bham.ac.uk/~mhe/papers/omniscient-journal-revised.pdf
Thanks,
Oliver
My attempt:
4>19-3x
Subtract 19 from both sides:
-15 > -3x
Divide both sides by -3:
5 > x
Switch sides (change sign):
x < 5
! But Maths Genie tells me the answer is x>5
Where have I gone wrong?
Why quarks numbers of the three colors where exactly equal in the initial quarks-gluons plasma? Seeing the Avogadro number, even a 10E-20 additional excess in one color should give a lot of free quark tracks in bubble chamber.
Suppose i have a term like this one (repeated indices are being summed)
$$x = \psi^a C_{ab} \psi^b$$
Such that ##C_{ab} = - C_{ba}##, and ##\{\psi^a,\psi^b\}=0##. How do i evaluate the derivative of this term with respect to ##\psi_r##?
I mean, my attempt g oes to here
$$\frac{\partial...
Hello guys,
I am refreshing on complex numbers today; kindly see attached.
ok for part (a) this is a circle with centre ##(\sqrt{3}, -1)## with radius =##1## thus, we shall have,
The attempt at a solution:
I tried the normal method to find the determinant equal to 2j. I ended up with:
2j = -4yj -2xj -2j -x +y
then I tried to see if I had to factorize with j so I didn't turn the j^2 into -1 and ended up with 2 different options:
1) 0= y(-4j-j^2) -x(2j-1) -2j
2)...
Let ##\Omega## here be ##\Omega=\sqrt{-u}##, in which it is not difficult to realize that ##\Omega ## is real if ##u<0##; imaginary, if ##u>0##. Now, suppose further that ##u=(a-b)^2## with ##a<0## and ##b>0## real numbers. Bearing this in mind, I want to demonstrate that ##\Omega## is real. To...
Hello guys,
I am able to follow the working...but i needed some clarification. By rounding to the nearest integer...did they mean?
##z=1.2-1.4i## is rounded down to ##z=1-i##?
I can see from here they came up with simultaneous equation i.e
##(1-i) + (x+iy) = \dfrac{6}{5} - \dfrac{7i}{5}## to...
First I solved 4+j3, which I squared 4 and 3 to equal 16 and 9 then I added them to get 25 and then I got the square root of 25 = 5.
Then I plugged it back in to the equation.
[50/(5)(50)+100] x 150 to get 50/350x 150= 1/7(150)= 21.42. I've attached the correct answer.
Kurzesagt in a Nutshell said that the number of possible protein combinations the human body can have is 6.8 x 10^495. I asked GPT to multiple it by 20 million (which is the hypothetical number of possible alien civilizations in the Milky Way galaxy give or take). The chatbot gave me 1.36 x...
How should I write an account of prime numbers in arithmetic progressions? Assuming this account should be in the form of an essay of at least ## 500 ## words. Should I apply the formula ## a_{n}=3+4n ## for ## 0\leq n\leq 2 ##? Can anyone please provide any idea(s)?
I'm trying to code Newton Raphson's method for finding zeros. I realize that even if the solution is real, it's possible for guesses to be complex. For example:
$$y=\sqrt{x-6}-2$$
While 10 is a valid real root, for any guess less than 6, the result is complex.
I tried to run the code allowing...
https://www.wired.com/story/this-random-video-game-powers-quantum-entanglement-experiments/
I don't understand the principle of this experiment. The gamers produced random numbers, and what was done with these numbers then? Was the value like <S> in CHSH inequalities computed, and was it...
Good afternoon everyone,
I had a quick question on the correct way to report the answer of this problem.
I've solved it and came to the conclusion that the bag of gravel experiences no static friction, but the box experiences static friction with a magnitude equal to the weight of the bucket -...
TL;DR Summary: Eight Exact To Get 49
Please advise if I am wording this problem correctly and what are the solution (is there some equation for combinations )/ Answers:SET of numbers 1,2,3,4,5,6,7,8,9,10,11,12 months of the year.
Within the exact time frame of 10 years I MUST choose a number...
Here is my attempt(photo below), but somehow the solution in the textbook is z= 2 - (3/2)i, and mine is z=(-3/2) +2i.
Can someone please tell me where I am making a mistake? I suppose it's something with x being part of the real part of the 1st complex number and x being part of an imaginary...
The quantum number n determines the energy, and for each n the allowed values for the angular momentum quantum number are -(n-1),...,(n-1).
This doesn't seem resonable to me. Classically increasing the orbital angular momentum will result in an increase in the energy of the system. But why is it...
How can I solve a system of equations with complex numbers
2z+w=7i
zi+w=-1
I have tried substituting z with a+bi and I have tried substituting w=7i-2z but didn't get anything useful.
Edit: also, I've tried, multiplying lower eq. with -1 so that I can cancel w but I get stuck with 2z and zi and...
Question is indeed very short and clear, but answer will eliminate many doubts and misleading conclusions.
Theory predict downwash that will cause reduction in airflow angle,this we call effective airflow(airflow with lower angle compare at geometric AoA). Does wing of aircraft during flight...
My question is [following the example on the attachment which is apparently clear to me].
1. Are the numbers ##eπ## and ##e+ π## Transcendental?
2. Algebraic numbers can also be rational and not necessarily integers? is that correct?
My interest is only on part (a). Wah! been going round circles to try understand why the radius = ##2##. I know that the given sequence is both bounded and monotonic. I can state that its bounded above by ##1## and bounded below by ##0##. Now when it comes to the radius=##2##, i can also say...
https://www.scientificamerican.com/article/what-is-known-about-tachy/
How do tachyons have mass of a square root of a negative number? I thought this was not mathematically possible?
Is there a name for the union of {prime numbers} and {integers that are not powers of integers}?
For example, we would include 2, 3, 5, 7, 11... And also 6, 10, 12...
But we exclude 2^n, 3^n, ... and 6^n , 10^n , etc.
What are some interesting contexts where this set crops up?
The problem is as shown...all steps are pretty easy to follow. I need help on the highlighted part in red. How did they come to;
##z^4+8z^2+16-9z^2=0## or is it by manipulating ##-z^2= 8z^2-9z^2?## trial and error ...
I know that this expression evaluates to 1 when a is equal to 0. Also for when a is equal to 1/n when n is a positive number, but I'm confused about how to go about this in double precision?
how does capacitors and inductors cause phase difference between current and voltage? how does complex number come into play to explain the relation between phase of current and voltage?
Proof:
Let ## p ## be the prime divisor of two successive integers ## n^{2}+3 ## and ## (n+1)^{2}+3 ##.
Then ## p\mid [(n+1)^{2}+3-(n^{2}+3)]\implies p\mid (2n+1) ##.
Observe that ## p\mid (n^{2}+3) ## and ## p\mid (2n+1) ##.
Now we see that ## p\mid [(n^{2}+3)-3(2n+1)]\implies p\mid...
Hello,
I'm posting here since what follows is not about homework, but constitutes a personal research which underlies some more general questions.
As with the infamous "casus irreducibilis" (i.e. finding the real roots of a cubic function sometimes requires intermediate calculations with...
Suppose I have 2 collections of lists.
In the first collection the lists consists of random integers, with most (but not all) in the range 0-1000.
In the second collection the lists consist of integers calculated in the following way:
a. start with a random integer of similar range to the...
Hi PF!
Everyone knows that: $${\varphi }^2 - \varphi - 1 = 0$$ But guess what? $${\varphi}^3-2{\varphi}^2+1=0$$ Generalizing this for all n-bonacci numbers: $$x^{n+1}+1 = 2x^n$$ where ##x## is the n-bonacci number and ##n## is the degree of the polynomial that the n-bonacci number is a root of...
Was watching a youtube on Grahams number last night and its discoverer Ronald Graham. He talked about how many algorithms can calculate the ending digits of the number (its a power of 3) but the first digit is unknown. Guessing this is generally true of all incomputably large numbers that are...
I argue not. Let ##f:\mathbb{Q}\rightarrow\mathbb{R}## be defined s.t. ##f(r)=r^2##. Consider an increasing sequence of points, to be denoted as ##r_n##, that converges to ##\sqrt2##. It should be clear that ##\sqrt2\equiv\sup\{r_n\}_{n\in\mathbb{N}}##. Continuity defined in terms of sequences...
I wrote a program that implements the pattern and finds the primes automatically. It worked up to 70 million then it crashed because program holds data in RAM so it can be fixed. It found all the primes up to 70 million and found no exception. I won't explain the pattern because its so...
Hi PF!
I am wondering the differences between the discrete and continuous case for expected value of minimum of 3 integers uniformly distributed from 1 to 13 vs 3 reals from 1 to 13.
The real case is direct: ##F = ((x-1)/12)^3 \implies f = 3(x-1)/12)^2## for CDF ##F## and PDF ##f##. Thus the...
In [this post][1] user William Ryman asked what would happen if we try to build "complex numbers" with shapes other than circle or hyperbola in the role of a "unit circle".
[Here][2] I proposed three shapes that could work. The common principle behind them being
that if the unit curve is...
Closer to odd number implies ##|y/x - (2n+1)| < 1/2## for ##n = 0,1,2...##. Then
$$-\frac 1 2 < \frac y x - (2n+1) < \frac 1 2 \implies\\
y < (2n + 1.5)x,\\
y > (2n + 0.5)x$$
for each ##n##. We note ##x \in (0,1)## implies ##y## can be larger than 1 since the slope is greater than 1 (but we know...