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Questions tagged [rational-numbers]

Questions about numbers expressible as the quotient of two integers. For questions on determining whether a number is rational, use the (rationality-testing) tag instead.

A rational number is any number that can be expressed as the quotient or fraction $\frac pq$ of two integers, with the denominator $q$ not equal to zero. Since $q$ may be 1, every integer is a rational number. The set of all rational numbers is usually denoted by $\Bbb Q$; it was thus named in 1895 by Peano after quoziente, Italian for "quotient".

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defining the value of real numbers raised to rational exponents

if $b^{1/n}$ where $b$ is a real number and $n$ even is defined as the positive real solution to the equation $x^n=b,$ how did they suddenly decide to define the value of $b^{m/n}$ where $m$ and $n$ are integers (rational exponent). did they…
James Doherty
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Descartes' Rule of Signs, Factor and Rational Zeros Theorem

I was teaching this morning Descartes' Rule of Signs to my Precalculus class and I wrote this polynomial on the board: $f(x)=3x^5-2x^4+2x^3-3x^2+2x+1$ I found that there is 4 change of signs, therefore there is 4 positive real zeros, or 2 positive…
Joshua Bonet
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what does it mean to say that rational numbers are obtained "from unity"?

In Courant's Introduction to Calculus and Analysis (Introduction, page 2) he says that: [The rational numbers] are all obtained from unity by using the "rational operations" of calculation, namely, addition, subtraction, multiplication, and…
Stephen
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Multiplying Rational Expressions

I'm stuck on this equation. These are the steps I've done so far: And after this I'm stuck. 1 - 8? It makes no sense. I have no clue about what I did wrong. Any help is greatly appreciated.
Grimestock
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Rational root coefficient

I saw this question in my exam recently, If a, b, c are distinct rational roots of $x^3+ax^2+bx+c=0$, find the values of a, b, c. Can someone give me a hint or answer? I tried factoring it and then expand it all out, but it doesn't seem very…
blastzit
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What subsets of rationals have been defined where each element equals the sum of a sequence of numbers?

In particular, I'm interested in the rationals that result from adding a finite sequence of consecutive integer powers of two. Has it been studied somewhere? Update This formalized example might clarify the…
nightcod3r
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Convert rational number in $\frac {p}{q}$ form

Convert rational number in $\frac {p}{q}$ form $0.40\bar 7$ (here bar is over $7$). solution: By solving I got the answer $367/900$ by multiplying by $10$ My friends are getting answer $4037/9900$ by multiplying by $100$ Are they correct $?$
rst
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How do I write the opposite of a rational number?

Write the opposite of each rational number A)$ 9$ B)$-17.6$ C) $6.12 $ D) $-7 \frac{5}{7 }$ Some one please help! I am not doing very good in Math I'm in grade 9 and I'm struggling I would highly appreciate if someone would help me out. Thanks…
Sally
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Rational Zeros Theorem, get the possible solutions

In my book there is a section about rational numbers. The first example show that:6^(1/3) cannot represent a rational number. In the proof it says that the possible solutions are ±1,±2,±3,±6. Since that none of these numbers satisfies the equation…
BioShock
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What is a rational number? What is the quotient of two integers?

How can we form an idea of the result of the operation of dividing a by b with a and b integers and b not equal to zero?
Peter
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How can we prove that every rational number has a terminating or periodic decimal form?

The title says it all. I'm aware of the proof of the converse of my statement, but how do I go on about proving this. Any help would be appreciated.
sayantankhan
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How to solve this Ratio problem

I am preparing for an olympiad and came across the following question in a workbook. There is answer but no explanation: The ratio between the number of passengers travelling by Ist and IInd class between two railway stations is 1:50, whereas the…
Govind Balaji
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Finding Rational numbers

Please help with the following question: Find rational numbers a and b such that: $$\left(7 + 5\sqrt2\right)^{\frac13} = a + b \sqrt2$$ Thank you
Charlie
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Confusion over whether $1/2$ and $2/4$ should be counted as two different elements of $\mathbb{Q}^{+}$ or not

The instruction for proving that the set of rational numbers, $\mathbb{Q}^{+}$, is countable, is as follows: First assume each $x\in \mathbb{Q}^{+}$ is in reduced form. Construct a one-to-one function $p: \omega \times \omega \rightarrow…
Tran Khanh
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