the geometric progression involves the assumption that 1/10n --> 0 as n --> ?. i am thinking the arrows imply a limit, because no matter how big the denominator is, there will still be a number, but for most purposes it is almost equal to zero. thus i do not think it is an exact value.
But, a the limit has a well-defined, exact value so the sum is an exact value. Part of the problem with the name "limit" is that from the connotations of the word limit, some students believe that limits represent approximations. But, limits are not approximations, they are exact values. A whole branch of mathematics, analysis, is built on the fact that limits can have well-defined, exact values.
what is the difference between natural numbers and real numbers? are natural numbers a subset of real numbers?
Here's a brief overview of some of the common numerical structures used in mathematics:
The real numbers, denoted R
, while intuitively easy to define and understand, have a very complex formal definitions in mathematics. Therefore, I think it will suffice to say that the real numbers represent any finite (i.e. not infinite) quantity you can think of that does not involve i (sqrt of -1). In a physical sense, the real numbers are measurements you can obtain from an instrument. For example, lets say you are measuring the displacement of a particle. You can obtain values which are negative or positive. The values can be integers (e.g. 1, 3, 104
), non-integers (e.g. 1.5, 4.32), and even numbers whose decimal representations do not end (e.g. pi, e, sqrt(2)). The real numbers form a mathematical structure called a field.
The real numbers are a subset of the complex numbers, denoted C
. Complex numbers involve, so-called imaginary numbers (square roots of negative numbers). They are of the form:
a + bi
where a and b are real numbers and i denotes sqrt(-1). Note that for b=0, you recover the real numbers. Like the reals, the complex numbers form a field.
There are various subsets of real numbers. Natural numbers, denoted N
, are "counting numbers." In a physical sense, a natural number would be what one would answer to the question "how many people are in this room?" or "how many molecules occupy 1L?" Since you cannot have half of a person or a fraction of a molecule, the answers must be natural number such as 1, 10, 504, 6.02x10^23. Similarly, you cannot have a negative number of people or molecules, so the natural numbers are restricted to positive numbers. Some people consider 0 a natural number, but some people do not.
The natural numbers are a subset of the integers, denoted by Z
. Integers are better known as "whole numbers" and they are formed by taking the natural numbers and their additive inverses (i.e. their negatives). The integers form a mathematical structure known as a ring.
From the definition of integers, we can define other subsets of real numbers, such as the rational numbers, denoted Q
. Rational numbers are any number which can be represented by a ratio of two integers. Therefore, the rational numbers include the integers (ratio of an integer with 1), and all fractions. The rationals also form a field. The irrational numbers (which I will denote Qc
are any number which is not rational (i.e. has no representation as a ratio of two integers). Irrational numbers are numbers with never-ending, non-repeating decimal representations, such as pi, e, and sqrt(2). Note: not all numbers which never-ending decimals are irrational -- for example, 0.333... = 1/3 is rational despite having a never-ending decimal. In fact, any number with a repeating, never-ending decimal is rational.
Together the rational numbers and irrational numbers partition the real numbers. In summary:N
where < denotes subset. Note that there are other, more complex structures which can be built on top of these basic numerical structures. For example, on top of a field structure, you can build a vector space. For example, the Cartesian coordinate plane (x,y) is the vector space R2
, because every element of R2
can be described by two real numbers.
For more information you can see the Wikipedia article: http://en.wikipedia.org/wiki/Number