Unformatted text preview: tions, positive and negative. The union of the set
of r ational numbers and the set of ir r ational numbers is called the set of real
irr
ir
numbers R .
−2
S −3 −
2 −3
−
2 −
1 −1
−
2 0 1
−
2 1 3
−
2 π e S2 2 3 proper
Real number
umbers:
COMPLETENESS pr oper ty of the Real n umber s:
Corresponding to every point on the number line we have a unique real number
number
point
and vice versa. Between any two real numbers on the number line each and every
point corresponds to a real number. The set of real numbers R is COMPLETE. The
real number line is smooth and CONTINUOUS. There are no gaps, breaks or bumps.
This Real number line we call the xaxis
xaxis.
We now see the connection between the set R of real numbers in Algebra and a
straight line (a continuous set of points) in Geometry. Each point xi on the xaxis
continuous
contin
points
corresponds to a Real number and vice versa. And when we say x1 < x2 for x1, x2 ∈ R
the picture from the Geometry point of view is :
ALGEBRA
GEOMETRY
set R
≡
xaxis
x1 x2
x1 < x2
13 4. TENDS T O and LIMIT
In elementary geometry you learned the definition or meaning of a point
point.
You also know how to name or label a point
point.
In Calculus a point on the number line or xaxis corresponds to a real number.
Sometimes we know its exact value, e.g. 2 . Sometimes we will know only the
approximate value. However, we may denote it by a special name or label
or symbol, e.g. S2 , e, π . R egardless of knowing the exact value or not we
know which point we are talking about.
In Calculus we prefer to use the word instant rather than point We speak of
point.
the instant 2, or the instant S2 , and so on. In Calculus we have another
way to define an instant Before we do this we need to know what an
instant.
infinitesimal is.
In order to give a formal definition of an inf initesimal we need to use two
infinitesimal
concepts: TENDS TO and LIMIT.
TENDS TO:
Let a = 0.5 and x = 0.4, 0.49, 0.499, 0.4999, . . . progr essively.
It is clear th...
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This note was uploaded on 11/29/2012 for the course PHYSICS 105 taught by Professor Tamerdoğan during the Fall '09 term at Middle East Technical University.
 Fall '09
 TAMERDOğAN

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