56 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

CHANGE FROM ONE SCALE TO ANOTHER

123. Suppose that a number a is written in one scale and
that we wish to change to another. Three methods are
available.

Let 7 be the radix of the old notation and a,, ay, @s, -+, @n
the digits when the number is exprest in this notation.
Then a = a7 + @pyr™ 1+ ++« + aor? + a1 + a,
= Qn? + @n1)? + Qn2)? + +++ + a2)7 + a)r 4 ao
§ 101.

It is usually an easy matter to express the old radix and
the old digits in the new notation. Then by computing in the
new notation according to either of the above formulas we will
have the number @ exprest in the new notation.

For example, to express 4398('f) in the scale whose radix
is 7.
We have 4398(f) =4 X P*4+3 X Y2 +9X ¥ + 8
= EX P 3 X P+ XY +8
Now: in thescaleof 7,8 = 11,9 = 12, ' = 13.
Hence, operating in that scale as follows,

Y= 13 Y= 13
42 36
13 12
2= 202 9 XY =156
13
606
202

P3 = 2626