Number Systems Help
Introduction to Number Systems
Calculus is one of the most important parts of mathematics. It is fundamental to all of modern science. How could one part of mathematics be of such central importance? It is because calculus gives us the tools to study rates of change and motion. All analytical subjects, from biology to physics to chemistry to engineering to mathematics, involve studying quantities that are growing or shrinking or moving—in other words, they are changing . Astronomers study the motions of the planets, chemists study the interaction of substances, physicists study the interactions of physical objects. All of these involve change and motion.
In order to study calculus effectively, you must be familiar with cartesian geometry, with trigonometry, and with functions. We will spend this article reviewing the essential ideas.
The number systems that we use in calculus are the natural numbers, the integers, the rational numbers, and the real numbers . Let us describe each of these:
- The natural numbers are the system of positive counting numbers 1, 2, 3, .... We denote the set of all natural numbers by .
- The integers are the positive and negative whole numbers and zero: ..., −3, −2, −1, 0, 1, 2, 3, .... We denote the set of all integers by .
- The rational numbers are quotients of integers. Any number of the form p/q , with p, q and q ≠ 0, is a rational number. We say that p/q and r/s represent the same rational number precisely when ps = qr . Of course you know that in displayed mathematics we write fractions in this way:
- The real numbers are the set of all decimals, both terminating and non-terminating. This set is rather sophisticated, and bears a little discussion. A decimal number of the form x = 3.16792 is actually a rational number, for it represents
A decimal number of the form
m = 4.27519191919 ...,
with a group of digits that repeats itself interminably, is also a rational number. To see this, notice that
100 · m = 427.519191919 ...
and therefore we may subtract:
100 m = 427.519191919 ...
m = 4.275191919 ...
Subtracting, we see that
99 m = 423.244
So, as we asserted, m is a rational number or quotient of integers.
The third kind of decimal number is one which has a non-terminating decimal expansion that does not keep repeating . An example is 3.14159265 .... This is the decimal expansion for the number that we ordinarily call . Such a number is irrational , that is, it cannot be expressed as the quotient of two integers.
In summary: There are three types of real numbers: (i) terminating decimals, (ii) non-terminating decimals that repeat, (iii) non-terminating decimals that do not repeat. Types (i) and (ii) are rational numbers. Type (iii) are irrational numbers.
You Try It : What type of real number is 3.41287548754875 ... ? Can you express this number in more compact form?
Practice problems for this concept can be found at: Calculus Basics Practice Test.
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