SCALES OF NOTATION 45

103. We have the following corollary to the radix theorem.
Th. Every positiv integer is a sum of powers of 2 (including
the zeroth power of 2, which is 1).()

Ex.1. 19=1+ 2+ 2¢

Ex.2. 31 =14 240820
3. 32=25

Ex.4. 33=1-42°
5
6,

Ex. 5. Express as sums of powers of 2 the numbers 1, 2, 3,
.-+, 16, 87, 96, 436, 783.
QUANTICS

104. Def. If we have given a set of finite numbers ay, a1,
s, +*+, Gn, Where 7 is some given positiv or zero integer, and
suppose that x may represent any one of some set of finite
numbers, then the set of values computed from the values of
x by the formula

anx"® + @px™t + oo+ ax? + a1x + ao,

n+1
or Z (an_k+1xn—k+1) ,

k=1
is called a quantic.

We sometimes speak of the expression itself as a quantic;
AnX", GnaX™ L, <o+, A2x?, @1, Qo are called its terms; a,, @n,

-+, as, a1, ao are called the coefficients of these terms, also
called the coefficients of the quantic.

The value of the quantic is determind by the value of x,
that is, it is obtaind from x by an operation (a unary operation,
since there is only one operand x).

For example, the value of the quantic x 4+ 7, in which
n=1,a,=a = 1and a, = 7, is obtaind from the value of
x by adding 7 to the value of x; the value of the quantic 3x?,
in which n = 2, @, = a2 = 3, a1 = 0, ap = 0, is obtaind from
that of x by the operation of squaring x and multiplying the
result by 3; the value of the quantic 5x* — x% + 2, in which
n=23a,=as=>5a=—1a =0,a = 2,is obtaind from

(*) See J. Bowden, The Russian Peasant Method of Multiplication, The
Mathematics Teacher, Vol. V, No. 1, Sep. 1912,