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anemone
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Let $x$ be an irrational number. Show that there are integers $m$ and $n$ such that $\dfrac{1}{2555}<mx+n<\dfrac{1}{2012}$.
ALI ALI said:$x$ is irrational then $\mathbb{Z} + x \mathbb{Z}$ is dense
An irrational number is a real number that cannot be expressed as a ratio of two integers. This means that it cannot be written as a fraction with a finite number of digits after the decimal point.
Solving a challenge involving irrational numbers usually involves using approximations or estimation techniques, as it is not possible to find the exact value of an irrational number. This may include using decimal approximations or rounding to a certain number of significant figures.
Yes, irrational numbers can be negative. Examples of negative irrational numbers include -√2 and -π.
The main difference between rational and irrational numbers is that rational numbers can be expressed as a ratio of two integers, while irrational numbers cannot. Additionally, rational numbers have a finite or repeating decimal representation, while irrational numbers have an infinite, non-repeating decimal representation.
Irrational numbers have many real-life applications, including in geometry, physics, and engineering. For example, the value of π is used in calculations involving circles and spheres, and the value of √2 is used in the Pythagorean theorem.