(2)
Figure I
96 IV. Real Numbers
Figure 2
Alternatively, the methods of calculus could be used to determine circular arc lengths and hence radian measure and the wrapping function 8/ . In either way, this function plays a central role in analytic trigonometry (§9).
The development of Euclidean geometry from its foundations in the axioms also provides a treatment of the magnitudes of areas (of triangles, rectangles, circles, etc.) and a comparison of these magnitudes with those measured along the real line. This is carried out in Hilbert's
Foundations of Geometry.
Thus one might say that the axiomatic foundations of geometry must provide for the comparison and reduction to a single scale of the various types of geometric magnitudes-including also volumes.The same must be done in any completed system of non-Euclidean geometry.
Measurements can also reduce other types of physical magnitudes to a geometric form. Thus weights on a balance are compared with numbers (the number of weights needed to achieve balance), while the weights measured on other instruments give the magnitude of a weight by a pointer position along some geometric scale. A thermometer does the same for the magnitudes of temperature, while a clock hand reduces time to angular magnitudes.
To summarize: Many comparisons or measurements of various magni
tudes can be reduced to the single scale provided by the real numbers.
And in those cases where a single real number does not suffice to measure a magnitude, it is often fitting to use several such numbers-as when the size of a plane figure is given by its width and its height, or of a solid figure by width, height, and depth. In other words, that scale of real numbers has many different realizations.
3 . Manipulations of Magnitudes 97
3. Manipulations of Magnitudes
When two objects are put together, side by side or end to end, the magni
tude of this combination is just the sum of the two separate magnitudes.
This operation, called
addition,
arises for all sorts of magnitudes-for distance, weight, time, height, area, and the like. Geometrically, the addition of two segments consists in laying off one segment after the other along the line. This geometric operation corresponds exactly to the arithmetic operation of adding numbers.
A second operation on magnitudes, that of multiplication, is suggested both by the multiplication of numbers and by geometric formulas; thus the area of a rectangle is obtained by multiplying its base by its height. A complete geometric description of the multiplication of segments requires more than one line in the plane. Thus to multiply two positive linear mag
nitudes
x
andy
one may represent them on two intersecting linear scales:Segment OA and then A B on the first line with measures 1 and
x,
respectively, and then OA ' with measure
y
on a second ray from O. Then drawing (Figure 1) BB ' parallel to AA ' constructs similar triangles OAA ' and OBB ', while the proportionality theorem for similar triangles makes OA ' IOA = A 'B ' lAB and hence shows that A 'B ' represents the magnitude
xy.
This may be regarded as the geometricdefinition
of multiplication of magnitudes.
Other types of magnitudes such as weight or time may more easily be multiplied first by whole numbers-thus to multiply the weight of a given item by three, take the combined weight of three such items. By division, this extends first to multiplication by rational numbers and then by con
tinuity to the multiplication of a weight by any number, rational or irra
tional. It is again remarkable that one gets the same operation of multipli
cation for
all
these types of magnitudes.To summarize: The "practical" operations of addition and multiplica
tion on various types of magnitudes lead to the algebraic operations of sums and product for the real numbers on the linear scale. The various rules for these manipulations of numbers were well known before they
Figure I
98
IV. Real Numbers were codified by axioms. We record the codification: The real numbers form an abelian group under addition, under addition and multiplication they form a commutative ring, moreover, one which is a field. Here acommutative ring
R is a set of elements (numbers) which is an abelian group under an operation of addition, which has an associative and commutative binary operation of multiplication with a unit (a number I such that
r
·l =r
for allr
in R) and in which both distributative laws hold :a(b
+c)
=ab
+ac (b
+c)a
=ba
+ca .
( I ) (Since multiplication is commutative, either one o f these two laws would suffice; it is a curious observation, repeated in other axiom systems, that here a single axiom, the distributive law, suffices to tie together the separately axiomatized additive and multiplicative structures.) Finally, afield
is a commutative ring in which every equationxa
= I witha =1=
0 has a solutionx,
necessarily unique. The remarkable fact is thatall
thealgebraic
rules for the manipulation of sums and products of real numbers follow from this simple list of axioms-a list far more perspicuous than the axioms of plane geometry, reflecting in part the fact that the line (the scale) is geometrically simpler than the plane, and that the axioms for the line more strictly separate the algebraic structure from the order structure (§4).For the immediate purposes of this chapter, we could have formulated these axioms just as properties of the real numbers. With the more gen
eral terms (ring and field) defined above, we can note that there are already at hand other examples of fields (the rational numbers Q and Z/(p), the integers modulo a prime). There are many more examples of commutative rings, including the system Z of integers and Zn , the integers modulo any
n.
More generally, aring
is defined by dropping the commutative law
ab
=ba
in our definition of commutative ring, but retainingboth
distributive laws (I). Adivision ring
is a ring in which every equationxa
= I anday
= I fora =1=
0 has a solutionx
ory.
These general notions, involving a possibly non-commutative multiplication, are recorded here for convenience. They do not belong here in the order of ideas, which now should concern only the scale of reals and its remarkable properties.
4. Comparison of Magnitudes
Many practical observations about magnitudes amount to the determina
tion of which of two magnitudes is the greater. In geometry, the notion of betweenness provides for such a comparison of the magnitudes of two segments. On a line, the directed segment A B is less than the directed seg
ment A C if and only if B lies between A and C. If the real numbers b and
4. Comparison of Magnitudes 99
e
represent these segments A B and A C, then this defines the relationb < e (b
less thane)
for real numbers b ande.
The axioms appropriate to this relation "less than" can be derived (though we will not carry this derivation out) from the geometrical axioms for betweenness. They state that the real numbers form an"ordered field" .
Here a fieldF
is said to be (Note that here too there is just one axiom connecting order and addition, and just one axiom connecting order and multiplication.) Observe also that the rational numbers (as well as the reals) form an ordered field in this sense.These axioms provide for all the usual manipulations of inequalities. In particular, a real number