A naive question about irrational numbers

Thread starter la6ki
Start date Oct 20, 2012
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Irrational Irrational numbers Numbers

In summary, the conversation discusses the concept of using mathematics to describe reality and the limitations of this approach. It also touches on the idea of irrational numbers and their relation to physical measurements. The conclusion is that while mathematics may not give an exact answer in the physical world, it can provide a close approximation and be a useful tool for modeling and prediction.

Nov 6, 2012

Integral

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Every REAL measurement has associated with it error bounds. Generally you can take your error to be 1/2 of the smallest division on your measuring instrument.

So your 1 m rod by measurement is 1m +/- Δ, where the magnitude of Δ is determined by your measurement instrument. Note that in the real world a device which will measure 1m to +/-.1mm is a specialty device which will cost you a good sum of money.

No matter how you do the measurement there will always be uncertinaty. That is why you cannot have a real rod exactly 1m. It may be, you just cannot prove it.

Your measurement of the √2 length will have the same errors, so the fact that √2 is irrational is immaterial, its length is known only as good as your ability to measure. Measuring any length to more then 3 or 4 decimal places is essentially impossible for the man on the street.

Last edited: Nov 6, 2012

<h2>1. What are irrational numbers?</h2><p>Irrational numbers are real numbers that cannot be expressed as a ratio of two integers. They are non-repeating and non-terminating decimals, such as pi (3.141592...) and the square root of 2 (1.414213...).</p><h2>2. How are irrational numbers different from rational numbers?</h2><p>Rational numbers can be expressed as a ratio of two integers, while irrational numbers cannot. Additionally, rational numbers have either a finite or repeating decimal representation, while irrational numbers have non-repeating and non-terminating decimals.</p><h2>3. Are irrational numbers important in mathematics?</h2><p>Yes, irrational numbers play a crucial role in mathematics. They are used in various mathematical concepts, such as geometry, trigonometry, and calculus. They also have practical applications in fields like engineering and physics.</p><h2>4. Can irrational numbers be approximated?</h2><p>Yes, irrational numbers can be approximated by rational numbers. For example, the decimal representation of pi can be approximated by 22/7 or 3.14. However, these approximations are not exact and can only be used to a certain degree of accuracy.</p><h2>5. How do irrational numbers affect our daily lives?</h2><p>Irrational numbers are used in many practical applications, such as in measurements, calculations, and technology. For example, irrational numbers are used in GPS systems to accurately determine locations, and in computer graphics to create smooth curves and shapes.</p>

1. What are irrational numbers?

Irrational numbers are real numbers that cannot be expressed as a ratio of two integers. They are non-repeating and non-terminating decimals, such as pi (3.141592...) and the square root of 2 (1.414213...).

2. How are irrational numbers different from rational numbers?

Rational numbers can be expressed as a ratio of two integers, while irrational numbers cannot. Additionally, rational numbers have either a finite or repeating decimal representation, while irrational numbers have non-repeating and non-terminating decimals.

3. Are irrational numbers important in mathematics?

Yes, irrational numbers play a crucial role in mathematics. They are used in various mathematical concepts, such as geometry, trigonometry, and calculus. They also have practical applications in fields like engineering and physics.

4. Can irrational numbers be approximated?

Yes, irrational numbers can be approximated by rational numbers. For example, the decimal representation of pi can be approximated by 22/7 or 3.14. However, these approximations are not exact and can only be used to a certain degree of accuracy.

5. How do irrational numbers affect our daily lives?

Irrational numbers are used in many practical applications, such as in measurements, calculations, and technology. For example, irrational numbers are used in GPS systems to accurately determine locations, and in computer graphics to create smooth curves and shapes.