The Evolution of the

R E A L  N U M B E R S

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WHATEVER IS CONTINUOUS we call a magnitude.  Some of the magnitudes of geometry are length, area, and volume (or solid). Some of the magnitudes of physics are speed, force, and time. A magnitude is what we measure rather than count.  That is the difference between a magnitude and a natural number.

Problem 1.   

a)  6 pencils could be divided into which parts?

Halves, thirds, and sixths. Pencils are discrete.

b)  6 meters could be divided into which parts?

Any parts. Meters are continuous.


A square, a circle, a cube

Geometry is the study of figures.  A square is a figure.  A circle and a cube are figures.  A figure is whatever has a boundary.  The boundary of a plane (flat) figure is the magnitude length.  (We do not mean length as opposed to width.  The sides of a square are lengths.  The circumference of a circle is a length.)  The space enclosed by the boundary of a plane figure -- the figure itself -- is area.  Length and area are different kinds of magnitudes.

Problem 2.   What kind of a magnitude -- length, area, or volume (solid)  -- is each of the following?

a)  A plane figure, such as a circle.   Area

b)  The boundary of a plane figure.   Length

  c)   A side of a triangle.    Length   d)    A cube.    Volume

e)  The boundary -- the faces -- of a cube.   Area

f)  The edge of a cube.   Length

  g)   A triangle.    Area   h)   A pyramid.    Volume
  i)   A sphere.   Volume   j)   The surface of a sphere.    Area

k)   The equator of a sphere.   Length


There is one number, clearly, that must be associated with these strokes:

Six strokes

But is there one number that must be associated with this length, as

A length

its measure?

No.  It will depend on the unit of measure.  For if we measure in inches, we will get one number, while if we measure in meters, we will get another.  Unlike counting, measurements are not absolute.

How do we "measure" AB?  We take a unit of measure -- 1 inch, 1 meter, 1 mile -- and then name the ratio of AB to that unit.

1 and a length AB

Every measurement implies a ratio to the unit of measure. Every measurement implies a proportion.

A magnitude is to the unit magnitude (of the same kind) as
A number is to 1.

For if we say that AB is 3 meters, that means

AB : 1 meter = 3 : 1.

Problem 3.   What proportion is implied by each of the following?

a)  The length L is 5 miles.

a)   L : 1 mile = 5 : 1

b)  The length L is 7.62 cm.

b)   L : 1 cm = 7.62 : 1

c)  The weight W is 5½ pounds.

c)   W : 1 pound = 5½ : 1

d)  The area A is 2.71 square meters.

d)   A : 1 square meter = 2.71 : 1

e)  The volume V is .035 cubic centimeters.

e)   V : 1 cc = .035 : 1

Problem 4.   In the previous problem, each measurement is a rational number of units.  Therefore, express each ratio as a ratio of natural numbers (Lesson 7.).

a)  The length L is 5 miles.   5 : 1

b)  The length L is 7.62 cm.

b)   7.62 : 1 = 762 : 100

c)  The weight W is 5½ pounds.

c)   5½ : 1 = 11/2 : 1 = 11 : 2

d)  The area A is 2.7 square meters.

d)   2.7 : 1 = 27 : 10

e)  The volume V is .035 cc.

e)   .035 : 1 = 35 : 1000

Problem 5.

a)    What is "the ratio of two natural numbers;" that is, what is the relationship that natural numbers have to one another?
  One number is either a multiple of another, a part of it, or parts of it.
b)    Do you expect that magnitudes (of the same kind) will have the same ratio as two natural numbers?
  Do you?
c)    In particular, if 1 centimeter is the unit of length, do you expect that every length will be a rational number of centimeters?
  Do you?
End of Section

Next Topic:  Common measure

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