Questions on basic arithmetic, e.g. addition, subtraction, multiplication, division, powers, radicals, etc.

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1answer
46 views

How many numbers from 1 to 2^n have '11' as a substring in binary representation? [duplicate]

For example: say $n = 2$. The numbers from $1$ to $2^2$ are $1, 2, 3, 4$. i.e. $1, 10, 11, 100$ in binary. So the result is $1$, because only one number i.e. $3$ is there such that it has ...
0
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1answer
68 views

Formula of MIPS (million instructions per second)

Could you please help me to understand the mathematics behind MIPS rating formula? The performance of a CPU (processor) can be measured in MIPS. The formula for MIPS is: $$MIPS = \frac{Instruction \ ...
1
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8answers
211 views

Is $4 \times 6$ defined as $4 + 4 + 4 + 4 + 4 + 4$ or $6 + 6 + 6 + 6$? [closed]

There are long debates among Indonesian netizens about this http://www.globalindonesianvoices.com/15785/is-4x6-the-same-as-6x4-this-primary-school-math-made-controversy-in-social-media/
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vote
1answer
35 views

Hypothetical scenario with economics

You have been assigned to purchase a new molding machine. One vendor offered a machine that will cost $200,000$, with an estimated installation of $10,000$. The machine has an expected life of $10$ ...
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2answers
101 views

How is $\frac{1-x}{x^2-1}=\frac{1}{x+1}$?

When integrating $\int \frac{1-x}{x^2-1} dx$ Maple rewrote it as $-\int\frac{1}{x+1}dx$ How is $\frac{1-x}{x^2-1}=\frac{1}{x+1}$?
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4answers
77 views

How can I show that $(x-1)(x^2-1)$ divides the polynomial $(x^n-1)(x^{n+1}-1)$? [closed]

How can I show that $(x-1)(x^2-1)$ divides the polynomial $(x^n-1)(x^{n+1}-1)$?
1
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1answer
33 views

$f \in \Sigma_n^1 \iff f \in \Pi_n^1$ in an analytical hierarchy

The proposition 1.7 in Higher Recursion Theory by Sacks states $f \in \Sigma_n^1 \iff f \in \Pi_n^1$ with the proof: Since $f$ is a function, then, $f(x)=y \iff \forall z. [y \neq z \implies f(x) ...
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0answers
29 views

Proving the additive inverse of a sum is the sum of the additive inverses

I have this proposition to prove: For all $m, n, \in\mathbb Z$: $-(m + n) = (-m) + (-n)$ Proof: \begin{align*} -(m + n) + (m + n) &= 0 + 0\\ -(m + n) + (m + n) &= (-m) + m + (-n) + n\\ (-m) + ...
2
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1answer
436 views

How to find antilog with simple calculator?

I know how to find log to base $10$ using simple calculator: say if you want to find log of $12$ you can do as blow: Step 1: $13$ times $\sqrt{\star} \implies 1.00030338$; Step 2: subtract 1: ...
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2answers
986 views

Why are linear functions linear?

I always thought linear functions need to satisfy $$f(x+y)=f(x)+f(y).$$ I am a tad confused now, consider $f(x)=2x+3$. $f(1)=5$, $f(2)=7$, $f(1+2)=f(3)=9 \neq f(1)+f(2)$ which was what I thought ...
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3answers
94 views

Prove the uniqueness of subtraction

I have to prove this proposition: Given $m,n \in\mathbb Z$, there exists one and only one $x \in\mathbb Z$ such that $m + x = n$. So, just to be sure: I am given an equation and asked to first prove ...
2
votes
1answer
82 views

When is $(p - 2)! \equiv 1 (\bmod p)$

I want to show when the following is true for $p$ a prime number. $(p - 2)! \equiv 1 \pmod p$. Could someone help me prove this? It worked for $p = 2$, $p = 3$, $p = 5$, so I believe it may work for ...
-1
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1answer
64 views

What is the difference between decomposing a number and partitioning a number?

I see that the question has been answered for SETS in "The Decomposition VS. The Partition of a set" but I would like good definitions that distinguish between these terms when used for NUMBERS. I ...
0
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1answer
27 views

Arithmetic progression sum equation

I have faced an equation that I just cannot solve. I know that it must be solved using geometric progression sum forumla, but I don't know how to find the difference. Maybe you could help me? :) $$ ...
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2answers
56 views

How to perform arithmetic (4-functions) on Von Neumann numbers?

Using the Von Neumann representation of the non-negative integers, where the empty set corresponds to zero, and the successor function is defined as the function on a set that returns the union of the ...
0
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1answer
38 views

Converting from twos complement to decimal?

I am currently reading a textbook and I can't seem to understand what the examples in the book did. I do believe it is an error with the book, but if not can someone explain? How come there is no ...
1
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0answers
39 views

Real Roots of$ x^{10} + a_9x^9 + a_8x^8 +…+a_1x -5$

The equation $ x^{10} + a_9x^9 + a_8x^8 +.....+a_1x - 5 =0 $ where $a_i $are real numbers. Then 1. Must have one real root of multiplicity one Must have atleast two real roots (distinct or ...
1
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1answer
29 views

Relation between tolerance and correct decimal places

If I want to find an answer that is correct to say 10 decimal places, do I use a tolerance of $10^{-10}$ or $10^{-11}$ for example? In other words, is the tolerance to be used equal 10^ (- number of ...
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2answers
30 views

Inverse pair equidistant from 1

Say you pick a number $x$, like $\frac 43$. Its inverse is of course $\frac 34$. $x$ is a distance of $\frac 13$ away from 1, and its inverse is a distance of $\frac 14$ away from 1. Is there any ...
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1answer
59 views

Proving Decimal Representation

Prove that if the decimal representation of a nonnegative integer n ends in 5 or 0 then 5 | n. (Hint: As a first step show that if the decimal representation of a nonnegative n integer ends in d0 then ...
0
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1answer
43 views

Adding/Subtracting in binary and hexadecimal number systems?

I have two numbers(in decimal): M = 3892.74 N = 9341.65 I am trying to add and subtract them in binary numbers and then in hexadecimal numbers. I manage to ...
2
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1answer
44 views

Field Completions

Let $(K,v)$ be a number field with an absolute value. Denote $K_v$ to be a completion of $K$ and $\overline{K_v}$ to be an algebraic closure of $K_v$. Let $E$ be a finite extension of $K$. Every ...
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1answer
29 views

why are these two intuitive ways of understanding division equivalent?

The elementary school example of division, say 12:4 is saying that you have to share out 12 cookies to 4 kids. However, another (only slightly less intuitive) way would be to ask how many times 4 ...
2
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3answers
370 views

Finding product without working it out

What's the easy way to find the solution for the below problem without actually multiplying the numbers: $$(24 * 24) - (16 * 16)?$$ I tried multiplying the numbers but that is a long way. The ...
2
votes
0answers
34 views

Why have multiplicative operators precedence over additive operators?

Considering that addition is (in my understanding) a more basic operation than multiplication, would it not make sense to give it higher priority? That is to say, we would expect to encounter more ...
0
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1answer
27 views

Minimize g(x, y)=max{12-x,8-y} over x+y=10

Let $g (x, y)=\max\{12-x,8-y\}$. Find the minimum value of $g(x,y)$ as $(x, y)$ varies over the line $x+y=10$. 1.5 2.1 3.7 4.3 My attempt Without use of Lagrange method of multiplier , I see ...
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2answers
41 views
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0answers
13 views

Upper bound on function defined by induction involving divisions by two

Let $A=\lbrace (x,y)\in{\mathbb N}^2 | 0<x<y \rbrace$. For positive integers $u,v$ let $\rho(u,v)=(\textsf{min}(u,v),\textsf{max}(u,v))\in A$. It is easy to see that there is a unique map ...
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0answers
41 views

How many possibilities of sequences are there to have averages of adjecent two in the sequence will be as provided

Given: Sequence of Averages of adjecent two numbers in list i.e. (n-1) integers Output: how many non-decreasing sequences of n elements are possible? for example: Given: 2 4 Possibilities: 1> 0 4 ...
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2answers
94 views

Is $\Delta_0=\Delta_1$ in arithmetical hierarchy?

I have seen a definition (e.g. http://www.math.ubc.ca/~bwallace/ArithmeticalHierarchy.pdf) of an arithmetical hierarchy in computability starting with: "let $\Delta_0=\Sigma_0=\Pi_0$ be the set of all ...
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2answers
146 views

How to solve $\sqrt{9-4\sqrt{5}}=$?

Need some hints how to solve this: $\sqrt{9-4\sqrt{5}}=$ ? Thanks.
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2answers
52 views

What is it called and how it works?

If I borrow from two of my parents 50 dollars and spend 45 dollars at the store, I am left with 5. On my way home, I borrow to my friend 3, now I am left with just 2. I return home and give 1 to my ...
2
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4answers
81 views

find $z$ that satisfies $z^2=3+4i$

Super basic question but some reason either I'm not doing this right or something is wrong. The best route usually with these questions is to transform $3+4i$ to $re^{it}$ representation. Ok, so ...
0
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1answer
39 views

roots of equation $ e^{2x}\sin (2x) +7=0$

Let a,b,c be roots of equation $ e^{2x}\sin2x -7=0$ then roots of equation $ e^{2x}\sin2x +7=0$ lies between p and q where Both p and q $\in$(a, b) Both p and q $\in$(b, c) p $\in $(a, b) and q ...
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2answers
51 views

Problem approximating for a chemical problem:

Solve using suitable approximation by hand: $$s(s-\alpha)=5\times10^{-13} \\ \frac{\alpha}{(s-\alpha)(0.4-2\alpha)^2}=10^8$$ Now to solve this I tried assuming $s\approx\alpha\approx0.2$ and then ...
1
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1answer
18 views

Mathematical arithmetic rounding $1$

What is an example of $4$-digit arithmetic rounding: $ \text{ }\\ \text{a)}\\ 11.2468 = 11.25\\ \text{ }\\ \text{b)}\\ 0.25632 = 0.256\\ \text{or}\\ 0.25632=0.2563\\ $ Can you explain what is ...
3
votes
2answers
41 views

Solving an integer equation

Is it true that if: $x$, $y$, $z$ and $t$ are integers such that: $xz + 7yt = 0$ $and$ $yz + xt = 0$, then $x = y = 0$ $or$ $z = t = 0$? Why or why not? Unless I have miscalculations somewhere ...
0
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1answer
22 views

Arithmetic Time and Work related question

8 men can complet a piece of work in 20 days. 8 women can do the same work in 32 days. In how many days will 5 men and 8 women together will complete the same work? All I have is: Men's one day work ...
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4answers
233 views

Reversing the digits with a subtraction [closed]

How many 3-digits numbers possess the following property: After subtracting $297$ from such a number, we get a $3$-digit number consisting of the same digits in the reverse order.
0
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1answer
32 views

Why do the denominators of two fractions with numerators $1$ add up to a third fraction that have the special things below?

I found out that the denominators of two fractions with numerators $1$ add up to a third fraction that has the sum in the numerator and the product in the denominator. For example, ${1\over ...
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1answer
55 views

How to divide by negative numbers? [closed]

How to divide by negative numbers? Like what is the quotient and remainder when 24 is divided by $-5$ or $-24$ is divided by $5$?
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2answers
29 views

Why do consecutive triangular numbers in pairs like $6$ and $10$ always add up to a perfect square?

I was a bit surprised by this when I thought of it. Look here: $$001, 003, 006, 010, 015, 021, 028, 036, 045, 055$$$$004, 009, 016, 025, 036, 049, 064, 081, 100$$As you just saw, $15$ and $21$ add to ...
0
votes
1answer
18 views

Polynomial with integer values

I'm looking for a polynomial $P \in \mathbb Q[x]$ with $P(\mathbb Z) \subset \mathbb Z$, with $P \notin \mathbb Z[x]$ I found that $f_n := \frac{1}{n}x(x-1)(x-2)\dots(x-(n-1))$ is such an element. ...
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2answers
81 views

Can you recommend a book with techniques for solving hard algebra/arithmetic problems?

I'm a university student who never really studied maths in high school (I did the basic courses, but because I'm dyslexic I was to embarrassed to try the harder courses) now I'm getting back into it, ...
0
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2answers
33 views

what is the default order and direction of operation?

I have a division like this 16/8/4/2 what is the default way to do calculations when the bracket is not specified . Method 1 : Is it correct to go from right to left like [16/ (8 / { 4 / ...
0
votes
2answers
51 views

How many numbers end with $0,2,9$?

The question is very simple: How many positive integers from $900$ and down end with $0,2$ and $9$? I think it is either $270$ numbers or $271$ numbers, but I am not sure which one.
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1answer
47 views

What do I do after this?

The question goes.. In a recent marathon, $\frac{1}{10}$ of the participants finished in less than $3$ hours. $\frac{1}{3}$ of the remaining participants finished in $4$ hours or less. Out of those ...
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2answers
45 views

Splitting sum into two sums

Assuming that $f$ is a multiplicative arithmetic function. Let $n_1,n_2\in \mathbb{N}$ with $gcd(n_1,n_2)=1$. Consider the sum $$\large S=\sum_{a\mid n_1n_2}f(a).$$ Can I split the sum $S$ into two ...
2
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2answers
73 views

Is there another way to prove $(x-n)^2 = (n-x)^2$

Let's say $n$ is $4$. So, I came up with the solution below. $(x-4)^2 = (x-4)(x-4) = x^2 - 8x + 16$ $(4-x)^2 = (4-x)(4-x) = 16 - 8x + x^2 = x^2 - 8x + 16$ I was wondering if there is another way ...
1
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3answers
236 views

Convert 1-5 Grading Scale to 1-100 Grading System

I am creating a formula in Excel to convert 1-5 Grading Scale to 1-100 Grading System Suppose that I have the following table: 97-100 = 1.00 94 - 96 = 1.25 91-93 = 1.50 88-90 = 1.75 85-87 = 2.00 ...