7
MEASUREMENT:
GEOMETRY AND ARITHMETIC
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.
Geometry
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
Measurement
Clearly, there is one number that must be associated with these strokes:
But is there one number that must be associated with this length,
as its measure?
Obviously, not. For if we measure in inches, we will get one number, while if we measure in meters, we will get another. How then do numbers become associated with magnitudes? How do we "measure" AB?
We measure AB by taking a unit of measure -- 1 inch, 1 meter, 1 mile -- and forming the ratio of AB to that unit.
Every measurement indicates a ratio to the unit of measure, and hence implies a proportion.
A magnitude : Unit magnitude (of the same kind) = A number : 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. L : 1 mile = 5 : 1
b) The length L is 7.62 cm. L : 1 cm = 7.62 : 1
c) The weight W is 5½ pounds. W : 1 pound = 5½ : 1
d) The area A is 2.71 square meters. A : 1 square meter = 2.71 : 1
e) The volume V is .035 cubic centimeters. V : 1 cc = .035 : 1
Problem 4. In the previous problem, each measurement is a rational number of units. (Topic 6.) Therefore, express each ratio to the unit of measure as a ratio of natural numbers.
a) The length L is 5 miles. 5 : 1
b) The length L is 7.62 cm. 7.62 : 1 = 762 : 100
c) The weight W is 5½ pounds. 5½ : 1 = 11/2 : 1 = 11 : 2
d) The area A is 2.7 square meters. 2.7 : 1 = 27 : 10
e) The volume V is .035 cc. .035 : 1 = 35 : 1000
Problem 5.
a) What is "the ratio of two natural numbers;" that is, what is the
a) 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
b) ratio as two natural numbers?
Do you? In particular, do you expect
b) that every length will be a rational number of inches?
Do you?
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