Category Archives: High School Calculus

Pi and the Circle

Posted on October 8, 2011 by Guillermo Bautista | Leave a comment

Pi ( $\pi \approx 3.1416$ ) is probably the most popular among all the irrational numbers. It is seen in almost all fields of mathematics and it often appears in places that you least expect it to be (like in birthday party perhaps?).

$\pi$ is the ratio of the circumference ( $C$ ) of a circle and its diameter ( $d$ ). That is, if we take a circle of any size, measure its circumference, measure its centimeter and divide them*, then Continue reading →

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Posted in High School Calculus, High School Mathematics

Tagged approximation of pi, area of a circle, inscribed polygon, pi

Is 0.999… really equal to 1?

Posted on June 4, 2010 by Guillermo Bautista | 6 Comments

Introduction

Yes it is. 0.999… is equal to 1.

Before we begin our discussion, let me make a remark that the symbol “…” in the decimal 0.999… means that the there are infinitely many 9′s, or putting it in plain language, the decimal number has no end.

A Friendly Chat About Whether 0.999… = 1 | BetterExplained via kwout

For non-math persons, you will probably disagree with the equality, but there are many elementary proofs that could show it, some of which, I have shown below. A proof is a series of valid, logical and relevant arguments (see Introduction to Mathematical Proofs for details), that shows the truth or falsity of a statement.

Proof 1

$\frac{1}{3} = 0.333 \cdots$

$\frac{2}{3} = 0.666 \cdots$

$\frac{1}{3} + \frac{2}{3} = 0.333 \cdots + 0.666 \cdots$

$\frac{3}{3} =0.999 \cdots$

But $\frac{3}{3} = 1$ , therefore $1 =0.999 \cdots$

Proof 2

$\frac{1}{9} = 0.111 \cdots$
Multiplying both sides by 9 we have

$1 = 0.999 \cdots$

Proof 3

Let $x = 0.999 \cdots$

$10x = 9.999 \cdots$

$10x - x = 9.999 \cdots - 0.9999 \cdots$

$9x = 9$

$x = 1$

Hence, $0.999 \cdots = 1$

Still in doubt?

Many will probably be reluctant in accepting the equality $1 = 0.999 \cdots$ because the representation is a bit counterintuitive. The said equality requires the notion of the real number system, a good grasp of the concept of limits, and knowledge on infinitesimals or calculus in general. If, for instance,you have already taken sequences (in calculus), you may think of the $0.999 \cdots$ as a sequence of real numbers $(0.9, 0.99, 0.999,\cdots)$ . Note that the sequence gets closer and closer to 1, and therefore, its limit is 1.

Infinite Geometric Sequence

My final attempt to convince you that $0.999 \cdots$ is indeed equal to $1$ is by the infinite geometric sequence. For the sake of brevity, in the remaining part of this article, we will simply use the term “infinite sequence” to refer to an infinite geometric sequence. We will use the concept of the sum of an infinite sequence, which is known as an infinite series, to show that $0.999 \cdots = 1$ .

One example of an infinite series is $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots$ . If you add its infinite number of terms, the answer is equal to 1. Again, this is counterintuitive.

How can addition of numbers with infinite number of terms have an exact (or a finite) answer?

There is a formula to get the sum of an infinite geometric sequence, but before we discuss the formula, let me give the geometric interpretation of the sum above. The sum $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots$ can be represented geometrically using a 1 unit by 1 unit square as shown below. If we divide the square into two, then we will have two rectangles, each of which has area $\frac{1}{2}$ square units. Dividing the other half into two, then we have three rectangles with areas $\frac{1}{2}$ , $\frac{1}{4}$ , $\frac{1}{4}$ square units. Dividing the one of the smaller rectangle into two, then we have four rectangles with areas $\frac{1}{2}$ , $\frac{1}{4}$ , $\frac{1}{8}$ , $\frac{1}{8}$ . Again, dividing one of the smallest rectangle into two, we have five rectangles with areas $\frac{1}{2}$ , $\frac{1}{4}$ , $\frac{1}{8}$ , $\frac{1}{16}$ , and $\frac{1}{16}$ Since this process can go on forever, the sum of all the areas of all the rectangles will equal to 1, which is the area of the original square.

Now that we have seen that an infinite series can have a finite sum, we will now show that $0.999 \cdots$ can be expressed as a finite sum by expressing it as an infinite series. The number $0.999 \cdots$ can be expressed as an infinite series $0.9 + 0.09 + 0.009 + \cdots$ . Converting it in fractional form, we have $\frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \cdots$ .

We have learned that the sum of the infinite series with first term $\displaystyle a_1$ and ratio $r$ is described by $\displaystyle\frac{a_1}{1-r}$ . Applying the formula to our series above, we have

$\displaystyle\frac{\frac{9}{10}}{1-\frac{1}{10}} = 1$

Therefore, the sum our infinite series is 1.

Implication

This implication of the equality $0.999 \cdots =1$ means that any rational number that is a non-repeating decimal can be expressed as a repeating decimal. Since $0.999 \cdots =1$ , it follows that $0.0999 \cdots =0.1, 0.00999 \cdots=0.01$ and so on. Hence, any decimal number maybe expressed as number + 0.00…01. For example, the decimal $4.7$ , can be expressed as $4.6 + 0.1 = 4.6 + 0.0999 \cdots = 4.6999 \cdots$ . The number $0.874$ can also be expressed as $0.873 + 0.001 = 0.873 + 0.000999 \cdots = 0.873999 \cdots$

Conclusion

Any of the four proofs above is actually sufficient to show that $0.999 \cdots = 1$ . Although this concept is quite hard to accept, we should remember that in mathematics, as long as the steps of operations or reasoning performed are valid and logical, the conclusion will be unquestionably valid.

There are many counterintuitive concepts in mathematics and the equality $0.999 \cdots = 1$ is only one of the many. In my post, Counting the Uncountable: A Glimpse at the Infinite, we have also encountered one: that the number of integers (negative, 0, positive) is equal to the number of counting numbers (positive integers) and we have shown it by one-to-one pairing. We have also shown that the number of counting numbers is the same as the number of rational numbers. Thus, we have shown that a subset can have the same element as the “supposed” bigger set. I guess that is what makes mathematics unique; intuitively, some concepts do not make sense, but by valid and logical reasoning, they perfectly do.

Notes:

You can find discussions about 0.999… = 1 here and here.
There is another good post about it here and here.

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Posted in Calculus and Analysis, High School Calculus

Tagged 0.999 equals 1, 0.999... does not equal 1, 0.999... exactly equals 1, 0.999...=1, concept of limits, counting numbers, does 0.999 1, infinite geometric sequence, infinite geometric series, limits and infinitesimals, mathematical proofs, rational numbers, real number system, repeating decimal, sequence of real numbers

The Definition of “Undefined”

Posted on April 21, 2010 by Guillermo Bautista | 9 Comments

In learning mathematics, we often encounter terms that are not always clear. Example of such term is the word undefined. What do we mean by undefined?

The word undefined may slightly differ in meaning depending on the context. In plain language, it means something which has no sensible meaning, or something that is ambiguous. For instance, during the time when the negative numbers were not yet invented, the numerical expression 5 – 8 has no meaning. In our time, we can say that 5 – 8 is undefined in the set of positive integers.

Below are some examples of the different contexts where the different meanings of “undefined” can be drawn.

Numbers

Square Root of Negative Numbers. The $\sqrt{-1}$ is undefined in the set of real numbers. This means that no real number exists that when multiply it by itself, the product is equal to -1. Note, however, that some operations maybe undefined under some sets, but defined in other sets. We know from high school mathematics that square root of – 1 equals i in the set of complex numbers.

Algebra

Division by 0. Since we do not know the answer if a number divided by 0, let us examine the quotient of numbers when divided by numbers close to 0. To make it simple, let us try 1/0.

Figure 1 - The value of the 1/x as x approaches 0.

As we can see, as x approaches 0 from the right, the quotient of 1/x is getting larger and larger. On the other hand, as x approaches 0 from the left, 1/x is getting smaller and smaller. As a consequence, there is no single number that 1/x approaches as x approaches 0. Therefore we can say that 1/0 is undefined.

Geometry

Intersection of two lines. In Euclidean geometry, if we talk about the intersection of two lines, we can have three cases: intersecting lines have one intersection, coinciding lines have infinitely many intersections, and parallel lines have no intersection.

Figure 2 - The three cases, in terms of intersection, two lines can be places on a plane.

We can say that if two lines are parallel, no intersection exists. Algebraically, the solution to the system of equations of the two lines is their intersection. Hence, the solution of the two systems of linear equations of parallel lines as graphs is undefined.

Calculus

Limits that Do not Exist. The limit of the function f(x) = 1/x as x approaches 0 cannot be determined. The value of the left hand limit is negative infinity and the value of the right hand limit is positive infinity. Since the left hand limit does not equal to the right hand limit, the limit of f(x) = 1/x does not exist or we can say that the limit is undefined.

Matrices

Matrices with Different Sizes. If A is a 2 by 2 matrix and B is a 3 x 3 matrix, then A + B has no meaning since five of the entries of matrix B have no corresponding entries in matrix A. We can say that the sum of matrix A and matrix B is not well-defined.

Sets

Intersections of Sets. If E is the set of even integers and O be the set of odd inteers, then there is no common value to both sets.

Figure 3 - The Venn Diagram of the intersection of even and odd numbers.

In set theory, we call the common values the intersection, and in this example, the intersection is the empty set. As a consequence, we can say that the intersection of set E and O is undefined.

***

Although the word “undefined” has different meanings depending on the context, by now, you would have realized that the phrases “does not exist”, “without sensible meaning” and “cannot be determined” are somewhat synonymous to it. If the result of an operation yields no value at all (note that 0 is a value and is not the same to no value), then it is more likely that it is undefined.