If you search images of the real number system on the internet, you will be surprised that there are a lot of incorrect real number system diagrams. One of the common incorrect diagrams is shown in the first figure below. If we interpret the diagram, all the numbers inside the the oval are real numbers. That is, our universal set is the set of real numbers.
Notice that there are numbers that are outside the rational number circle and also outside the irrational number circle that are real numbers (the solid bluish part). But we know that real number is either rational or irrational. Therefore, the numbers on the bluish part of the diagram outside the rational and irrational number circles do not exist. » Read more
Another representation of rational numbers aside from fractions is the decimal form. Every fraction has a decimal representation:
, , and .
Notice that some of the fractions above are terminating, while the others are repeating decimals. The fractions and have only one decimal place, while and have infinitely many (Can you see why?). Now, given a fraction, can we determine if it’s a terminating or non-terminating decimal without dividing?
First let us examine the characteristics of terminating decimals, say 0.125. The easiest way to convert this decimal into fraction is by dividing a whole number by a power of 10: » Read more
For you who is the greatest mathematician? This was a question asked to me by a kin, a freshman who is currently studying a mathematics related course. This question is probably asked by many others who are just starting studying mathematics or those who are just simply curious.
Asking who the greatest mathematician is like asking who the greatest singer is. Singers have different genre that it is nearly impossible to tell. Pop lovers would probably suggest that it was Michael Jackson, but classical singers would probably disagree and would suggest some names like Luciano Pavarotti.
I think, determining the greatest mathematician is even more complicated than determining the greatest singer. Mathematicians lived in different times and the maturity of mathematics at different times is enormously different. For example, during the time of Euclid, it takes a high-caliber mathematician to prove that the inscribed triangle in a circle containing its diameter is right, while they can be easily proved by eighth graders of the present time. Of course, we cannot claim that our eighth graders are better than or even at the same level as Euclid because mathematics has changed so much. Those who are only read by mathematicians during the time of Euclid are now taught in the elementary and high school levels. In addition, mathematicians study different fields and it is impossible to compare the level of difficulty or even to quantify the effect of their contributions. » Read more