Are all objects with irrational lengths measureable?

I was deleting old emails a while ago and I came across with questions from some students reading my blog.  I have answered quite a number of questions from middle school and high school students via Email and Facebook since this blog started. I think some are worth publishing here, so I’ll probably post one from time to time.  Below is the first Q & A in this series.

Question

Are all objects with irrational lengths measureable?

Answer

Yes. In principle, they are measureable.

The number line represents all real numbers. It contains all the rational and irrational numbers.  In fact, there is a one-to-one correspondence between the set of real numbers and the set of points on the number line.  This means that every real number has a corresponding point on the number line, and every point on the number line has a corresponding real number. Therefore, since we can locate every irrational number on the number line, we can find its distance from 0.  This distance represents the irrational length.

Why are Non-terminating, Repeating Decimals Rational

Last night, I received a Facebook message from a Grade 8 student asking why non-terminating repeating decimals are rational. I am posting the answer here for reference.

Rational numbers is closed under addition. That is, if we add two rational numbers, we are guaranteed that the sum is also a rational number. The proof of this is quite easy, so I leave it as an exercise for advanced high school students.

Before discussing non-terminating decimals, let me also note that terminating decimals are rational. I think this is quite obvious because terminating decimals can be converted to fractions (and fractions are rational). For example, 0.842 can be expressed as

\displaystyle\frac{842}{1000}.

Further, terminating decimals can be expressed as sum of fractions. For example, 0.842 can be expressed as

\frac{8}{10} + \frac{4}{100} + \frac{2}{1000}.

Since rational numbers is closed under addition, the sum of any number of fractions is also a fraction. This shows that all terminating decimals are fractions.  » Read more

Where is the Nobel Prize in Mathematics?

The Nobel Prize are prestigious awards given each year to individuals (as well as organizations) who have contributed significantly in cultural and scientific advances.

Alfred B. Nobel, the inventor of dynamite, bequeathed 31 million Swedish kronor in 1895 (about 250 million dollars in 2008) to fund the awards for achievements in Chemistry, Physics, Physiology and Medicine, Literature, and Peace. In 1901, the first set of awards were given, and in 1969, the Nobel Foundation established the Nobel Prize for Economics.

But did you ever wonder why there was no Nobel Prize of Mathematics? » Read more

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