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Questions tagged [real-numbers]

For questions about $\mathbb{R}$, the field of real numbers. Often used in conjunction with the real-analysis tag.

The field of real numbers, usually denoted by $\mathbb{R}$ or $\mathbf{R}$ is a field equipped with an order, which is complete with respect to that order. Moreover, it is the only ordered field which is complete (up to isomorphism). The real numbers are used as basis for measuring "length".

The real numbers can be classified in various ways: rational and irrational numbers; algebraic and transcendental numbers; computable and non-computable numbers; etc.

The real numbers carry a natural topology, which is generated by the order. The topology can be induced by a naturally arising complete metric. See more on Wikipedia.

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What are some extreme numbers that can be useful to have memorized for proprotionality?

In some areas you can sometimes encounter numbers so big or so small that they are nothing more than just a bunch of symbols that don't say much. In this situation knowing another extreme number somewhat close to what you're reading can be a great…
Daniel
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How to show that a function is bounded

Show that the following function $ f(x) = x sin \frac 1 x$ is bounded but it's derivative isn't. I don't know how to show this formally. I found the derivative which is $f'(x)=sin \frac 1x - \frac{cos{\frac 1x}}{x} $ but I'm stuck in showing how…
Arthur Lee
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Prove for $a > 1$, $n\in \mathbb{N}$, $a^\frac{1}{n} >1$

Prove for any $a > 1$, $n\in \mathbb{N}$, $a^\frac{1}{n} >1$ How do i prove this?
Little Rookie
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Are there special names for lower dimensional subspace of $\mathbb{R}^n$

For $n$-dimensional real number set $\mathbb{R}^n$, its $n-1$ dimensional subspaces are called hyperplanes. Are there special names for lower dimensional subspace of $\mathbb{R}^n$?
Guangliang
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A question of summation.

Let $X_1,X_2.....X_n$ be positive numbers such that $X_1+X_2.....+X_n=17$. Find the minimum of $X_1^2+X_2^2.....+X_n^2$. (Obviously in terms of n) ^_^
Yami Kanashi
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Normalizing decimal values into a range of 0 to 1

I want to normalize a range of decimals into a new range from 0 to 1. Lets say the maximum value of the range is 0.865400 and the minimum value of the range is 0.0004530. How can I convert values from the given range into a new range from 0 to 1 ?
Sanchit Agarwal
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$\lambda_1x_1 +\cdots+\lambda_kx_k = \max\{x_1,\cdots,x_k\}$ implies $x_1 = x_2 = \cdots = x_k$?

Is it true, and i case of it being true, how can I show that: (1) $\lambda_1x_1 +\cdots+\lambda_kx_k = \max\{x_1,\cdots,x_k\}$ (2) $\sum_{i=1}^{k}\lambda_i = 1$ and $\lambda_i \in (0 ,1] $ for $i = 1,\cdots, k$ Implies that $x_1 = x_2 =…
InterpolationKid
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Find the minimum value of expression involving real numbers

Let $n$ positive integer. Find the minimum value of expression: $$ E=max(\frac {x_1} {1+x_1},\frac {x_2} {1+x_1+x_2}, ... , \frac {x_n} {1+x_1+..+x_n})$$ where $x_1,x_2, .. , x_n$ are real not negative so that $x_1+x_2+ .. +x_n=1$ My try For…
user261263
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Is a number with two consecutive sets of differing infinite digits a real number?

Suppose I construct a number starting 0. (followed by an infinite number of digits that are 3) (followed by an infinite number of digits that are 1), like $0.\bar{3}\bar{1}$. Is this a real number? Can I do calculations with it? Is it real if I…
Math Man
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Proof of $\forall a \in \Bbb{R} -a = (-1)a$

Problem. Prove that $ \forall a \in \Bbb{R}: -a = (-1) a $. The teacher gave us a proof but I would like to see another :) Proof by contradiction Suppose that $ -a \neq (-1) a $. Then \begin{align} -a + a & \neq (-1) a + a, \\ a + (-a) & …
Jose Vega
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Defining $\mathbb{R}$ as a field

One of the definitions of real numbers I've encountered is "Dedekind complete totally ordered field". How to prove said field is unique? The definition seems to be not complete too, since rational numbers seem to be a Dedekind complete totally…
user285146
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Finding all real solutions to this equation

$$\frac{\sqrt{x} + \sqrt3}{\sqrt{x+\sqrt{x+\sqrt3}}} + \frac{\sqrt{x} - \sqrt3}{\sqrt{x-\sqrt{x-\sqrt3}}} = \sqrt3$$ I equalised with $0$, let $$u(x)= {\sqrt{x+\sqrt{x+\sqrt3}}}\cdot{\sqrt{x-\sqrt{x-\sqrt3}}}$$ Since $u(x)=0$ has no real solutions I…
diredragon
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Fractional power

By the definition of square root, we can take square root of $16$ as $\pm4$. But in problems, why do we take $16^{\frac12}$ as $4$? And also how can we say, if $a^m=b^n$ then $a=b^{m/n}$? Because when we take $x^2=a$, we solve it as…
Muzamil
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Reciprocal of a product with reciprocal: $(c \cdot d^{-1})^{-1}$ is equal to $c^{-1} \cdot d$

(My question is similar to this one at a high level, but I am looking for something more rigorous.) I have started into Michael Spivak's "Calculus" textbook. Problem 3 (v) on page 14 asks for a proof that "$\frac{a}{b} \big/\frac{c}{d} =…
sam.bishop
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How to define operations on Stevin's construction of real numbers?

I know Dedekind's and Cauchy's construction of real numbers, but what i find interesting is Stevin's construction. Let $p\in\mathbb{N}, p\ge 2$. We define $\mathbb{R}$ as a set of sequences $(a_0,a_1,a_2,\ldots)$ such that $a_0\in\mathbb{Z},…
Kulisty
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