College Mathematics Archives - Page 26 of 30

Rational and Irrational Numbers

May 5, 2010 GB College Mathematics, High School Algebra, High School Mathematics, Number Theory

The need of men to perform certain mathematical operations led to the birth of different types of numbers. People in the ancient times used only counting numbers to keep track of the number of their belongings such as animals. The concept of trade led to the invention of 0 and negative numbers. The need to divide led to the invention of rational numbers.

In this article, we are going to take a look at the characteristics of rational and irrational numbers.

Rational numbers are numbers of the form a/b where a, b are integers, and b not equal to zero. Rational numbers are called rational not because they are reasonable, but because they are a ratio of two integers. It is worthy to note the conditions in the definition. That is because not all fractions are rational numbers. For example, $\frac{2 \pi}{3 \pi}$ is a fraction, but it is only a rational number when simplified. From the definition, we can deduce that all integers are rational numbers since {…,-3, -2, -1, 0, 1, 2, 3,…} = {…, -3/1, -2/1, -1/1, 0/1, 2/1, 3/1, …}.

Geometric Interpretation of Rational and Irrational Numbers

In ancient times the Greeks, particularly the Pythagoreans, believed that all quantities are rational; that is, all quantities can be expressed as a ratio of two integers. Geometrically, this can be interpreted as follows. Given any two lengths, a unit length can be found that can measure the two lengths exactly without gaps or overlaps. In the example in Figure 1, we have two segments a and b, and we found a unit length that would fit exactly a whole number of times in both segments. The ratio of a:b is 7:6, or we can express it in a fraction that a is 7/6th of b.

Figure 1 – The division of segments a and b into unit lengths.

This belief, as most of us now know, was proven to be false. The Pythagoreans later discovered that given a square with length 1 unit, no unit length, however short, can be found to measure both the side of the square and its diagonal like what we have done above. They have concluded that the length of the diagonal cannot be expressed as the ratio of two integers and hence not rational. Today, numbers that are not rational are called irrational numbers. Hence, we define irrational numbers as numbers that cannot be expressed as a ratio of two integers.

Figure 2 – The square with length 1 unit has irrational diagonal.

Using the Pythagorean Theorem, we now know that the length of the diagonal of a square with side length 1 unit is equal to $\sqrt{2}$ . We have already discussed and proved that $\sqrt{2}$ is irrational.

The collection of all rational and irrational numbers is called real numbers. Geometrically, real numbers are represented by the real line as shown in Figure 3.

Figure 3 – The real number line.

Each real number can be represented by a point on the real number line and every point on the number line has a corresponding real number.

Another Representation of Rational and Irrational Numbers

Aside from fractions, we can also represent rational numbers with decimals. For example, 1/5 = 0.2 and 1/3 = 0.333…. Observe that 0.2 has a finite number of decimals while 0.333… has infinite. Irrational numbers can also be represented using decimals. They are the types of decimals that do not end and do not repeat.

Several irrational numbers are very popular, and we had been using them from elementary school to college. The irrational numbers $\pi$ , $e$ and $\phi$ are several of irrational numbers that we are acquainted with.

Figure 4 – The structure of the real number system.

From our discussion above, we can see that real numbers are divided into two main subsets – rational and irrational numbers.

Leave a comment irrational numbers, pythagorean theorem, rational numbers, real numbers

Division by Zero

May 3, 2010 GB Calculus and Analysis, College Mathematics, Questions and Quandaries

In studying mathematics, you have probably heard that division of zero is undefined. What does this mean?

Since we do not know exactly what is the answer when a number is divided by 0, it is probably reasonable for us to examine the quotient of a number that is divided by a number that is close to 0.

If we look at the number line, the numbers close to 0 are numbers numbers between –1 and 1.

Figure 1 – The number line showing the numbers close to 0.

For instance, several positive numbers close to 0 and less than 1 are 0.1, 0.01, 0.001 and so on. Similarly, negative numbers close to 0 but greater than – 1 are –0.1, -0.01, -0.001 and so on.

The table and the numbers below shows the quotient 1/x when 1 is divided by x, where the x’s are numbers close to 0.

Figure 2 – The value of 1/x as x approaches 0 from both sides.

In the graph, as x approaches 0 from the right (as x, where x are positive numbers, approach 0), the quotients of 1/x are getting larger and larger. On the other hand, as x approaches 0 from the left (as x, where x are negative numbers, approach 0), 1/x is getting smaller and smaller. Hence, there is no single number that 1/x approach as x approaches 0. For this reason, we say that 1/0 is undefined.

A simple analogy would also let us realize that allowing division by 0 will violate an important property of real numbers. For example 8/4 = 2 because 2 x 4 = 8. Assuming division of 0 is allowed. If 5/0 = n, then n x 0 = 5. Now, that violates the property of a real number that any number multiplied by 0 is equal to 0.

Since division by 0 yields an answer which is not defined, the said operation is not allowed.

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An extensive explanation about the Epsilon-Delta definition of limits

March 15, 2010 GB Calculus and Analysis, College Mathematics

One of the most important topics in elementary calculus is the $\epsilon-\delta$ definition of limits. The definition says that the $\lim_{x \to a} f(x)= L$ if and only if, for all $\epsilon > 0$ , there exists a $\delta >0$ such that if $|x-a| < \delta$ , then $|f(x) - L |<\epsilon$ . In this article, we are going to discuss what this definition means. Readers of this article must have knowledge about elementary calculus and the concept of limits.

Review of Limit Basics

Consider the function $f(x) = 2x$ . We have learned from elementary calculus that $\lim_{x \to 2} 2x = 4$ . Aside from algebraic computation, this is evident from the color-coded graph and the table shown in Figure 1. The yellow arrows in the graph and the values in the yellow cells in the table indicate that as the value of $x$ approaches $2$ from the left of the x-axis, the value of $f(x)$ approaches $4$ from below of the y-axis. On the other hand, the red arrows in the graph and the values in the red cells in the table indicate that as the value of $x$ approaches $2$ on from the right of the x-axis, the value of $f(x)$ approaches $4$ from above of the y-axis.

Figure 1 – The table and the graph showing the value of f(x) as x approaches 2 from both sides.

From the above discussion, it is noteworthy to mention three things:

We can get $f(x)$ as close to $4$ as we please by choosing an $x$ sufficiently close to $2$ . For example, I can set $x$ to $1.9999 \cdots99$ (with $100$ nines) to get an $f(x)$ very close to $4$ , which is $3.9999 \cdots 98$ ( $99$ nines).
No matter how small is the distance of $x$ from $2$ , a distance less than it may still be chosen. For example, if we choose the point which is very close to $2$ , say a point with coordinate $1.9999 \cdots 99$ with ( $100$ nines), we can still choose a value closer than this to $2$ . For instance, we can choose $1.9999 \cdots 99$ with $101$ nines. This can be repeated for every chosen distance.
Although $x$ can be very very close to $2$ , it does not necessarily mean that $x$ equals $2$ .

Now we go back to the definition of limits. In a specific example, the limit definition states that the $\lim_{x \to 2} 2x = 4$ if (and only if) for all distance (denoted by the Greek letter $\epsilon$ ) from $4$ along the y-axis (directly above or below $4$ ) – no matter how small – we can always find a certain distance (denoted by $\delta$ ) from $2$ along the x-axis (left or right of $2$ ) such that if $x$ is between $2 - \delta$ and $2 + \delta$ , then $f(x)$ would lie between $4 - \epsilon$ and $4 + \epsilon$ .

To give you a more concrete example, suppose we want the distance $\epsilon$ from $4$ , which is our limit, to be $0.1$ then the interval of our $f(x)$ is ( $4 -0.1, 4+ 0.1) = (3.9,4.1)$ . The definition of limit says that given a distance $\epsilon = 0.1$ , we can find a distance $\delta$ in the x-axis such that if $x$ is between $2 - \delta$ and $2 + \delta$ , we are sure that $f(x)$ is between $3.9$ and $4.1$ . We do not know the value of $\delta$ yet, but we will calculate it later.

Figure 2 – The epsilon-delta definition given epsilon = 0.1.

In Figure 2, $x$ is between $2 - \delta$ and $2 + \delta$ or $2 - \delta < x < 2 + \delta$ . Subtracting $2$ from all terms of the inequality, we have $- \delta < x - 2 < \delta$ . If you recall the definition of absolute value, this is precisely the same as $|x - 2| < \delta$ . The comparison among the notations is in Table 1.

Using the notations in the table, we can conclude that the following statements are equivalent:

Words: Given $\epsilon = 0.1$ , we can find a $\delta$ such that if $x$ is between $2 - \delta, 2 + \delta$ , then $f(x)$ is between $3.9$ and $4.1$ .
Set Notation: Given $\epsilon = 0.1$ , we can find a $\delta$ such that if $x \in (2 -\delta, 2 + \delta)$ , then $f(x) \in (3.9,4.1)$ .
Relational Operator: Given $\epsilon = 0.1$ , we can find a $\delta$ such that if $2 - \delta < x < 2 + \delta$ , then $3.9< f(x) < 4 .1$ .
Absolute Value: Given $\epsilon = 0.1$ , we can find a $\delta$ such that if $|x-2| < \delta$ , then $|f(x) - 4| < 0.1$ .

We have discussed that we can get $f(x)$ as close to $4$ as we please

by choosing an $x$ sufficiently close to $2$ . This is equivalent to choosing an extremely small $\epsilon$ , no matter how small, as long as $\epsilon>0$ . Our next task is to find the $\delta$ that corresponds to that $\epsilon$ .

Applying this definition to our example, we can say the $\lim_{x \to 2} 2x = 4$ if and only if, given $\epsilon > 0$ (any small distance above and below 4), we can find a $\delta > 0$ (any distance from x to the left and right of $2$ ) such that if $|x - 2| < \delta$ , then $|f(x) - 4| < \epsilon$ .

The Definition of a Limit of a Function

Now, notice that $4$ is the limit of the function as $x$ approaches $2$ . If we let the limit of a function be equal to $L$ and $a$ be the fixed value that $x$ approaches, then we can say that $\lim_{x \to a} f(x) = L$ if and only if, for any $\epsilon>0$ (any small distance above and below $L$ ), we can find a $\delta>0$ (any small distance from to the left and to the right of a) such that if $|x - a| < \delta$ then, $|f(x) - L| < \epsilon$ . And that is precisely, the definition of limits that we have stated in the first paragraph of this article.

Figure 3 – The epsilon-delta definition given any epsilon.

In mathematics, the phrase “for any” is the same as “for all” and is denoted by the symbol $\forall$ . In addition, the phrase “we can find” is also the same as “there exists” and is denoted by the symbol $\exists$ . So, rephrasing the definition above, we have $\lim_{x \to a} f(x) = L$ if and only if, $\forall \epsilon > 0, \exists \delta >0$ , such that if $|x - a| < \delta$ then, $|f(x) - L| < \epsilon$ . A much shorter version of this definition is the phrase $\lim_{x \to a} f(x) = L \iff, \forall \epsilon > 0, \exists \delta >0$ , such that $|x - a| < \delta \Rightarrow |f(x) - L| < \epsilon$ . The symbol $\iff$ stands for if and only if and the symbol $\Rightarrow$ is similiar to if-then. If $P$ and $Q$ are statements, the statement $P \Rightarrow Q$ is the same as the statement of the form “If $P$ then $Q$ “.

Finding a specific delta

We said that given any positive $\epsilon$ , we can find a specific $\delta$ , no matter how small our $\epsilon$ is. So let us try our first specific value $\epsilon = 0.1$ .

From the definition, we have $\lim_{x \to 2} 2x = 4$ if and only if, given $\epsilon> 0$ (any small distance above and below 4), $\exists \delta >0$ such that if $|x -2| < \delta$ then, $|f(x) - 4| < \epsilon$ .

Now $|f(x) - 4| = |2x - 4| < \epsilon$ . This implies that
$|2||x-2| < \epsilon$ which implies that $2|x - 2| <0.1$ . Simplifying, we have $|x- 2| < 0.05$ . This means that our $x$ should be between $1.95$ and $2.05$ to be sure that our $f(x)$ is between $3.9$ and $4.1$ . This is shown in Figure 4.

Figure 4 – The table showing some of the values of epsilon and delta satisfying the definition of limit of 2x as x approaches 2.

Now, let $\epsilon = 0.05$ . This means that our interval is $(3.95,4.05)$ . Now $|f(x) - 4| = |2x - 4| < 0.05$ . Thus, $|2||x -2| < \epsilon$ which implies that $2|x - 2| <0.05$ . Solving, we have $|x - 2| < 0.025$ . This means that our $x$ should be between $1.975$ and $2.025$ to be sure that our $f(x)$ is between $3.95$ and $4.05$ . There are only two examples above, but the definition tells us that we can choose any $\epsilon > 0$ so let us generalize our statement by doing so.

Now $|f(x) - 4| = |2x - 4| < \epsilon$ . This results to $|2||x -2| < \epsilon$ which implies that $2|x - 2| <\epsilon$ . Solving, we have $|x - 2| < \epsilon/2$ . From the condition above, $|x - 2| < \delta$ so we can let $\delta = \epsilon/2$ .

This means given any $\epsilon$ , we just let our $\delta$ equal to $\epsilon/2$ and we are sure that if $x$ is between $2-\delta/2$ and $2+\delta$ to be sure that our $f(x)$ is between $4 - \epsilon$ and $4 - \epsilon$ .

In the next calculus post, we are going to discuss the strategies on how to get $\delta$ given an arbitrary $\epsilon$ value, so keep posted.

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23 comments absolute value, concept of limits, definition of limits, epsilon-delta definition, epsilon-delta explanation, epsilon-delta proof, relational operators, representation of limits, set notation

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