What is Arithmetic: Definition and 476 Discussions
Arithmetic (from the Greek ἀριθμός arithmos, 'number' and τική [τέχνη], tiké [téchne], 'art' or 'craft') is a branch of mathematics that consists of the study of numbers, especially concerning the properties of the traditional operations on them—addition, subtraction, multiplication, division, exponentiation and extraction of roots. Arithmetic is an elementary part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry, and analysis. The terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory, and are sometimes still used to refer to a wider part of number theory.
This is the question I read in a Math quizzes book:
- you have the first 13 integers from 1 to 13
- someone gives you a code made of at most 4 digits (0 to 9)
- you decipher the code and tell him exactly which numbers he wanted from the list each number can be picked only once
For example.
case...
We have come to accept that Infinity times two is infinity. In the sense of 'size' we use to think about everyday numbers, the rules of arithmetic with infinities seem like nonsense. For example, consider the computable number
$$0.100100100100100....$$
In the decimal expansion, there are...
So I was thinking about arithmetic, geometric and harmonic means when I had a thought. Let's say we have a curve y = x^2. We want to find the AM of the points on the curve between x=1 and x=2 i.e. y = 1 and y = 4. To make thing easier, we'll start with just the endpoints and keep adding...
>>> from numpy import exp, pi
>>> exp(1j*pi)
(-1+1.2246467991473532e-16j)
The fact that the imaginary part of this is not zero is wrecking a fourier collocation scheme for a nonlinear PDE with periodic boundary conditions: the coefficient corresponding to the Nyquist frequency, which should be...
For the first part, I have used a= 2, b = eps/2, c = eps/2 which I believe works and I have tested it in MATLAB however haven't had any luck reproducing the second part in MATLAB with any numbers. Any hints? Thanks
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?
Hi,
Suppose I am given a sequence of special symbols and I want to produce the next sequence of symbols according to certain rules that transform one symbol into one or more symbols.
They are 10 symbols in total, say; A, B, C, D, E, F, G, H, I, J
The rules are:
A transformed into B and it is...
Hey,...
is that correct to say that "The null part added to itself will always remain the same as itself, namely null.""
In arithmetic, 3 * 0 = 0 + 0 + 0 = 0;
It is therefore healthy to extrapolate as follows:∞-1 * 0 = 0 + 0 + 0 +... + 0 + 0 + 0 + 0 = 0
∞ * 0 = 0 + 0 + 0 +... + 0 + 0 + 0...
My attempt;
The terms in the arithmetic sequence are ;##[ a, a+2d,a+5d]##.
It follows that;
Common ratio ##r=\dfrac{a+2d}{a}=\dfrac{a+5d}{a+2d}##
##⇒ar+2rd=a+2d+3d##
##ar+2rd=ar+3d##
##ar+2rd-ar=3d##
##2rd=3d##
##r=\dfrac{3d}{2d}=\dfrac{3}{2}##
The solution given on the textbook is...
Proof:
Suppose ## a\equiv b \mod n ##.
Then ## n\mid (a-b)\implies a-b=kn ## for some ## k\in\mathbb{Z} ##.
Since ## m\mid n ##, it follows that ## n=mp ## for some ## p\in\mathbb{Z} ##.
Note that ## a-b=k(mp)\implies a-b=mq ## where ## q=kp ## is an integer.
Thus ## m\mid (a-b) ##.
Therefore...
I try to list all the possible sequences:
1 2 3
1 3 5
1 4 7
2 3 4
2 4 6
2 5 8
3 4 5
3 5 7
4 5 6
4 6 8
5 6 7
6 7 8
I get 12 possible outcomes, so the probability is ##\frac{12 \times 3!}{8^3}=\frac{9}{64}##
But the answer key is ##\frac{5}{32}## . Where is my mistake? Thanks
https://www.feynmanlectures.caltech.edu/I_10.html
https://www.feynmanlectures.caltech.edu/I_09.html
Using Mathematical approach we can describe the motion of a falling body whose gravity is 32 m/s^2. Analysis shows that this is simply ##s-s_0=ut+1/2at^2##. Similarly we can describe the motion of...
Could someone explain why ##[a][x_0]=[c]\iff ax_0\equiv c\, (mod\, m)##?
My instructor said it came from the definition of congruence class. But I am not convinced.
How do I build functions by using Arithmetic Sequence, Geometric Sequence, Harmonic Sequence?
Is it possible to create all the possible function by using these sequences?
Thanks!
https://en.wikipedia.org/wiki/Klein%27s_Encyclopedia_of_Mathematical_Sciences
Originals are in German or French, the Japanese version cut all the historical content :(
Do you think that some day we will see this published in English?
Size is big, 20k pages, but it cannot be more interesting I...
The Fundamental Theorem of Arithmetic essentially states that any positive whole number n can be written as:
##n = p_1^{a_1} \cdot p_2^{a_2} \cdot p_3^{a_3} \cdot \dots##
where ##p_1##, ##p_2##, ##p_3##, etc. are all the primes, and ##a_1##, ##a_2##, ##a_3##, etc. are non-negative integers...
Let $a_1,a_2,\,\cdots,\,a_{2n}$ be an arithmetic progression of positive real numbers with common difference $d$. Let
(1) $a_1^2+a_3^2+\cdots+a_{2n-1}^2=x$
(2) $a_2^2+a_4^2+\cdots+a_{2n}^2=y$
(3) $a_n+a_{n+1}=z$
Express $d$ in terms of $x,\,y,\,z,\,n$.
Well, I am having a little difficulty knowing how to approach finding a solution to this problem. I am aware that in an arithmetic progression the first term is a and there is a constant common difference defined as d=un+1-un
Expanding the binomial given...
Summary:: Sequences, Progressions
Hello. I have been Given the following exercise, Let (a1, a2, ... an, ..., a2n) be an arithmetic progression such that the sum of the last n terms is equal to three times the sum of the first n terms. Determine the sum of the first 10 terms as a function of...
Question 1; Method 1
If the sum of the first four terms is 139 then S4=139
139=1/2(4)(2a+(4-1)d)
139=2(2a+3d)
139=4a+6d----- [1]
The part of this question that is confusing is the "the sum of the next four terms is 115".
Would this mean that S8=S4+115=139+115=254?
In which case...
The fundamental theorem of arithmetic applies to prime factorizations of whole numbers. Can this theorem also correctly be invoked for all rational numbers? For example, if we take the number 3.25, it can be expressed as 13/4. This can be expressed as 13/2 x 1/2. This cannot be broken...
I am thinking of taking modular of 13 to both sides of equation. So it will become
22x^2 is 11 mod 13.
And try all the values from x equal to zero to x equal to 11.
Is their easier way to solve it
I find most of the errors I make are related to arithmetic or using the wrong sign. (Note: This applies to mainly algebra and trigonometry for me.)
I've tried writing slower and neater, and have been structuring my operations linearly on the paper step by step.
Do any of you have some tried...
When people graduate and have their degrees in engineering or physics or mathematics or what they may have done, some of these people will use some mathematics, very often which is some-what complicated (or not) arithmetic. Why will some people choose to strictly avoid using variables in the...
Let x1, ..., x25 be such positive integers that x1⋅x2⋅ ... ⋅x25 = x1 + x2 + ... + x25. What is the maximum possible value of the largest of numbers x1, x2, ..., x25?
Here is a "problem" that is "befudding" folks on the internet. To the less mathematically mature, it might be vexing, but to us mathematically mature, we should have a clear comprehensive meme so that stupid things like this don't happen...
When they give reason for multiplication the negative numbers leading to positive number, they base on distribute law.But why the distribute law in algebra is correct like in arithmetic?(e.g why -5(8-6)=-5.8+-5.-6?).In abstract algebra they use distribute law as axiom.But in elementary algebra...
Hello,
I have a simple question and am hoping someone can help. I have a power plant that loses 0.5% production capacity per year for 25 years. When working with the plant capacity in terms of percentages, year 0 is defined as 100% capacity. For year 1 and each subsequent years, is it correct...
First, I know the answer: 301. I thought (despite the injunction that the problem is to be done without number theory) that the Chinese Remainder Theorem might be of help (if I would use a subset which contained only relatively primes), but that didn't get very far. I also tried to spot a...
##132,289≡1973* 67 + 98##
##1973≡98*20+13##
##98≡13*7+7##
##13≡7*1+6##
##7≡6*1+1##
now in reverse my attempt is as follows,
##1≡7-6##
## 1≡7-(13-7)##
##1≡2*7-(1973-20*132,289+1340*1973)##
##1≡2*7-(1341*1973-20*132,289##
which is correct but my interest is in finding the inverse of 1973 help?
A gumball machine contains 232323 green gumballs, 525252 red gumballs, 343434blue gumballs, 616161 yellow gumballs, and 303030 pink gumballs
What percentage are red ,do you add up
the elements and put it over the total?
Has anybody ever heard of this? I learned about it in a discrete math class in grad school, and I've never heard of it anywhere else !?
For example, logical disjunction (OR) and set-theoretic UNION are isomorphic in this sense:
0 OR 0 = 0.
{0} UNION {0} = {0}.
Similarly, logical AND & set...
In Peter Smith's Godel book, 2 conditions are proven of several that make Q "order adequate"
O2: For any n, Q ⊢ ∀x ({x=0 v x=1 v...v x=n} → x≤n)
03: For any n, Q ⊢ ∀x (x≤ n → {x=0 v x=1 v...v x=n})
O3 is proved by induction. O2 is not. It would appear as if induction would be required in...
Hi,
I am self studying induction and came across the following problem. I am stuck on how to proceed (I need to use induction, I know there is a direct proof). My proof attempt is as follows:
Let ## P (m) ## be the proposition that:
$$ \sum_{i = m + 1}^{n} i = \frac{(n - m)(n + m + 1)}{2} $$...
Homework Statement
[/B]
Weak Induction:
If (i) ##S(1)## holds, and (ii) for every ##k \geq 1(S(k) \Rightarrow S(k+1)##. Then ##\forall n \geq 1##, ##S(n)## holds.
Strong Induction:
If (i) ##S(k)## is true and (ii) ##\forall m\geq k [S(k) \land \cdots \land S(m)]\Rightarrow S(m+1)##. Then for...
If a,b, c, are in G.P and $\log_ba, \log_cb,\log_ac$ are in A.P. I want to find the common difference of A.P.
Answer:
After doing some computations, I stuck here. $\frac{2(\log a+\log r)}{\log a+2\log r}=\frac{2(\log a)^2+3\log r\log a +2(\log r)^2}{(\log a)^2+\log r\log a}$
How to proceed...
Greg Bernhardt submitted a new blog post
AVX-512 Assembly Programming: Opmask Registers for Conditional Arithmetic Conclusion
Continue reading the Original Blog Post.
Greg Bernhardt submitted a new blog post
AVX-512 Assembly Programming: Opmask Registers for Conditional Arithmetic
Continue reading the Original Blog Post.
Homework Statement
Let a and b be integers, and let m be a positive integer. Then a ≡ b (mod m) if and only
if a mod m = b mod m.
Homework EquationsThe Attempt at a Solution
By definition a ≡ b (mod m) => m| (a-b)
mx = a -b => mx + b = a => b = a mod m
b = a - mx => b = m(-x) + a => a = b...
NOTE:This is not a homework question! This is just a topic that I like very much,but don’t have the programming ability to do many of them.That’s why I post this thread.
C++ is a language without built-in big integer calculation functions,so building ones that can do such job is a great way to...
Calculation of probability with arithmetic mean of random variables
There are 4 people, each of whom has one deck of cards with 500 cards that are numbered from 1 to 500 with no duplicates.
Each person draws a card from his deck and I would like to calculate the probability of the event that...
Homework Statement
In an AP, sum of first n terms is equal to m and sum of first m terms is equal to n. Then, find the sum of first (m-n) terms in terms of m and n, assuming m>n.
Homework Equations
Sum of an AP: n/2 * {2a+ (n-1)d}
The Attempt at a Solution
We get two equations:
m= n/2 * {2a+...
The Beer-Lambert law gives the intensity of monochromatic light as a function of depth ##z## in the form of an exponential attenuation:
$$I(z)=I_{0}e^{-\gamma z},$$
where ##\gamma## is the wavelength-dependent attenuation coefficient.
However, if two different wavelengths are present...
Homework Statement
I'm making a program to display what I have posted in the image and my program needs to look just like it.
The idea is to have an employee enter information such as their pay and hours worked. Then taxes get calculated and it shows a net amount that the employee has made for...
Sir Michael Atiyah just gave a livestreamed talk claiming to prove the Riemann hypothesis. But it turns out that this is part of a larger research program in which he also claims to have an apriori calculation of the fine-structure constant and possibly other physical constants.
Atiyah is 89...