MATH 221 Numbers and Functions Assignment

1. What is a number?

1.1. Different kinds of numbers.

The simplest numbers are the positive integers
1, 2, 3, 4, · · ·
the number zero
0,
and the negative integers
· · · , ?4, ?3, ?2, ?1.
Together these form the integers or “whole numbers.”

Next, there are the numbers you get by dividing one whole number by another (nonzero) whole number.
These are the so called fractions or rational numbers such as
1/2, 1/3, 2/3, 1/4, 2/4, 3/4,4/3, · ·
By definition, any whole number is a rational number (in particular zero is a rational number.)

You can add, subtract, multiply and divide any pair of rational numbers and the result will again be a rational number (provided you don’t try to divide by zero). One day in middle school you were told that there are other numbers besides the rational numbers, and the first example of such a number is the square root of two. It has been known ever since the time of the Greeks that no rational number exists whose square is exactly 2, i.e. you can’t find a fraction m/n such that
(m/n)^2 = 2, i.e. m^2 = 2n^2.

Nevertheless, if you compute x2 for some values of x between 1 and 2, and check if you get more or less than 2, then it looks like there should be some number x between 1.4 and 1.5 whose square is exactly 2. So, we assume that there is such a number, and we call it the square root of 2, written as ? 2. This raises several questions. How do we know there really is a number between 1.4 and 1.5 for which x2 = 2? How many other such numbers are we going to assume into existence? Do these new numbers obey the same algebra rules (like a + b = b + a) as the rational numbers? If we knew precisely what these numbers (like ?2) were then we could perhaps answer such questions. It turns out to be rather difficult to give a precise description of what a number is, and in this course we won’t try to get anywhere near the bottom of this issue. Instead we will think of numbers as “infinite decimal expansions” as follows.

One can represent certain fractions as decimal fractions, e.g.
279 /25 = 1116/ 100 = 11.16.

Not all fractions can be represented as decimal fractions. For instance, expanding into a decimal fraction leads to an unending decimal fraction
1/3 = 0.333 333 333 333 333 · · ·

It is impossible to write the complete decimal expansion of because it contains infinitely many digits.

But we can describe the expansion: each digit is a three. An electronic calculator, which always represents numbers as finite decimal numbers, can never hold the number exactly.

Every fraction can be written as a decimal fraction which may or may not be finite. If the decimal expansion doesn’t end, then it must repeat. For instance,
1/7 = 0.142857 142857 142857 142857 . . .

Conversely, any infinite repeating decimal expansion represents a rational number.

A real number is specified by a possibly unending decimal expansion. For instance, ? 2 = 1.414 213 562 373 095 048 801 688 724 209 698 078 569 671 875 376 9 . . . Of course you can never write all the digits in the decimal expansion, so you only write the first few digits and hide the others behind dots. To give a precise description of a real number (such as ?2) you have to explain how you could in principle compute as many digits in the expansion as you would like.

During the next three semesters of calculus we will not go into the details of how this should be done.

1.2. A reason to believe in ?2.

The Pythagorean theorem says that the hypotenuse of a right triangle with sides 1 and 1 must be a line segment of length ?2. In middle or high school you learned something similar to the following geometric construction of a line segment whose length is ?2. Take a square with side of length 1, and construct a new square one of whose sides is the diagonal of the first square. The figure you get consists of 5 triangles of equal area and by counting triangles you see that the larger square has exactly twice the area of the smaller square. Therefore the diagonal of the smaller square, being the side of the larger square, is ? 2 as long as the side of the smaller square.

Why are real numbers called real? All the numbers we will use in this first semester of calculus are “real numbers.” At some point (in 2nd semester calculus) it becomes useful to assume that there is a number whose square is ?1. No real number has this property since the square of any real number is positive, so it was decided to call this new imagined number “imaginary” and to refer to the numbers we already have (rationals, ? 2-like things) as “real.”

1.3. The real number line and intervals.

It is customary to visualize the real numbers as points on a straight line. We imagine a line, and choose one point on this line, which we call the origin. We also decide which direction we call “left” and hence which we call “right.” Some draw the number line vertically and use the words “up” and “down.”

To plot any real number x one marks off a distance x from the origin, to the right (up) if x > 0, to the left (down) if x < 0>

The distance along the number line between two numbers x and y is |x ? y|. In particular, the distance is never a negative number.

lues of x but not for others. In modern abstract mathematics a collection of real numbers (or any other kind of mathematical objects) is called a set. Below are some examples of sets of real numbers. We will use the notation from these examples throughout this course.

The collection of all real numbers between two given real numbers form an interval. The following notation is used

• (a, b) is the set of all real numbers x which satisfy a < x>
• [a, b) is the set of all real numbers x which satisfy a ? x < b>
• (a, b] is the set of all real numbers x which satisfy a < x>
• [a, b] is the set of all real numbers x which satisfy a ? x ? b.

If the endpoint is not included then it may be ? or ??. E.g. (??, 2] is the interval of all real numbers (both positive and negative) which are ? 2.

1.4. Set notation.

A common way of describing a set is to say it is the collection of all real numbers which satisfy a certain condition. One uses this notation A = x | x satisfies this or that condition Most of the time we will use upper case letters in a calligraphic font to denote sets. (A,B,C,D, . . . )

For instance, the interval (a, b) can be described as (a, b) = x | a < x>

The set B = x | x2 ? 1 > 0 consists of all real numbers x for which x2 ? 1 > 0, i.e. it consists of all real numbers x for which either x > 1 or x < ?1 holds. This set consists of two parts: the interval (??, ?1) and the interval (1, ?).

You can try to draw a set of real numbers by drawing the number line and coloring the points belonging to that set red, or by marking them in some other way. Some sets can be very difficult to draw. For instance, C = x | x is a rational number can’t be accurately drawn. In this course we will try to avoid such sets.

Sets can also contain just a few numbers, like D = {1, 2, 3} which is the set containing the numbers one, two and three. Or the set E =  x | x3 ? 4x2 + 1 = 0 which consists of the solutions of the equation x3 ? 4x2 + 1 = 0. (There are three of them, but it is not easy to give a formula for the solutions.)

If A and B are two sets then the union of A and B is the set which contains all numbers that belong either to A or to B. The following notation is used A ? B = x | x belongs to A or to B or both.

Similarly, the intersection of two sets A and B is the set of numbers which belong to both sets. This notation is used:

A ? B =  x | x belongs to both A and B.

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