Math Home Search Dr. Fraction conversion to other bases What are the rules for converting fractions to binary and octal and vice versa? I've searched the archives and I only saw questions pertaining to whole numbers. Fraction conversion to other bases Hi, Celestine.
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Types of Infinity Most students have run across infinity at some point in time prior to a calculus class. However, when they have dealt with it, it was just a symbol used to represent a really, really large positive or really, really large negative number and that was the extent of it.
Once they get into a calculus class students are asked to do some basic algebra with infinity and this is where they get into trouble. This is not correct of course but may help with the discussion in this section. If you move into complex numbers for instance things can and do change.
When you add two non-zero numbers you get a new number. With infinity this is not true. With infinity you have the following. Likewise, you can add a negative number i.
Subtraction with negative infinity can also be dealt with in an intuitive way in most cases as well. A really, really large negative number minus any positive number, regardless of its size, is still a really, really large negative number.
Subtracting a negative number i. Multiplication can be dealt with fairly intuitively as well. Some forms of division can be dealt with intuitively as well. When we talk about division by infinity we are really talking about a limiting process in which the denominator is going towards infinity.
In other words, some infinities are larger than other infinities. However, with the subtraction and division cases listed above, it does matter as we will see.
Here is one way to think of this idea that some infinities are larger than others. This is a fairly dry and technical way to think of this and your calculus problems will probably never use this stuff, but it is a nice way of looking at this. For a much better and definitely more precise discussion see, http: So, pick any two integers completely at random.
Start at the smaller of the two and list, in increasing order, all the integers that come after that. Eventually we will reach the larger of the two integers that you picked. Because we could list all these integers between two randomly chosen integers we say that the integers are countably infinite.
Again, there is no real reason to actually do this, it is simply something that can be done if we should choose to do so. In general, a set of numbers is called countably infinite if we can find a way to list them all out. In a more precise mathematical setting this is generally done with a special kind of function called a bijection that associates each number in the set with exactly one of the positive integers.
To see some more details of this see the pdf given above. It can also be shown that the set of all fractions are also countably infinite, although this is a little harder to show and is not really the purpose of this discussion. To see a proof of this see the pdf given above. It has a very nice proof of this fact.
The following is similar to the proof given in the pdf above but was nice enough and easy enough I hope that I wanted to include it here. This means that there should be a way to list all of them out. The reason for going over this is the following.
An infinity that is uncountably infinite is significantly larger than an infinity that is only countably infinite. So, if we take the difference of two infinities we have a couple of possibilities.
Depending upon the context there might still have some ambiguity about just what the answer would be in this case, but that is a whole different topic.
We could also do something similar for quotients of infinities. Be careful when dealing with infinity.In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of metin2sell.com the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common, that is, there is no length ("the.
A terminating decimal does not repeat, such as or 1/4, as opposed to a repeating decimal, like or 1/3. Express the terminating decimal as the quotient of numbers.
The decimal is read as five-tenths. You can put this solution on YOUR website! How do you write the repeating decimal as a quotient of integers? N = Now since there are 2 digits that repeat multiply both sides by or N = Now write the first equation underneath the second equation, linng up the decimal points: N = Quotient of integers means dividing integers, so it is a fraction or a rational number all depending on how you look at it.
How to Write a Rational Number as the Quotient of Two Integers By Chris Deziel; Updated April 23, You can write the ratio between the two .
How do you write the repeating decimal as a quotient of integers? N = Now since there are 2 digits that repeat multiply both sides by or N =