340 Functions, graphs, and numbers
2 Write an infinite decimal which converges to an irrational number between
0.43211 and 0.43212.
3 Supposing that 0.31690416 and 0.31690444 converge to irra-
tional numbers, write a rational number that lies between them.
4 Supposing that a and b are different positive numbers, give a procedure
by which we can find a rational number x1 and an irrational number x2 that lie
between a and b.
5 For a long time before the advent of electronic computers, the base 10 of
the decimal system reigned supreme and most people thought that other bases
had only theoretical interest. Nowadays the base 2, which employs the two
binary bits 0 and I instead of the ten decimal digits 0, 1, 2, ., 9, is very impor-
tant. In the binary system, the left member of the formula
(1) bmb,n-i... b2blbo = bm2m + bm_12m-1 +. .. + b2222 + b12 + bo,
in which each bit bk is 0 or 1, abbreviates the right member. Thus the binary
representations of the first few positive integers are(2) 1, 10, 11, 100, 101, 110, 111, 1000, 1001,.Similarly,
L(3) (.b-1b-2b-3 ...)z = b=1 -f- 2 b=2-f- 22 b=$ 23 +b=4 24 -f- ...
where the subscript 2 in the left member informs us that the "point" is not a
"decimal point" but is a "binary point" and that each b is a binary bit. One
reason for importance of binary bits lies in the fact that one "state" such as
"light on" or "switch closed" or "true" can be represented by 1, while the oppo-
site "state" such as "light off" or "switch open" or "false" can be represented
by 0. Perhaps without knowing why, we can pick up useful ideas by solving
a few simple problems. Show that(a) (29)10 = (11101) 2 (b) (100) 2 = (4)10
(c) (100)10 = (1100100)2 (d) (416)10 = (110100000)2
(e) (10011)2 + (10110)2 = (101001)2 (f) ('is)10 = (0.00111)2
(g) (1010 = (0.01010101. ) 2
Remark: Many persons with substantial lacks of enthusiasm for adding, sub-
tracting, multiplying, and dividing with decimal digits can find genuine amuse-
ment in learning to make these manipulations with binary bits. Scientists need
never be bored because of lack of interesting things to do.
6 Inquisitive students may ask why we write(1) s=U1+us+u3+...
when the series converges to s. The answer lies partly in the fact that it is much
easier to write (1) than to write the statement that "s is the number to which the
series u1 + u2 + u3 +. .. converges" and partly in the fact that the method
of convergence which we have described is the simplest useful method for assigning
values to series. There are other methods that are both venerable and useful.
One of these is the method which is called the method of Abel (1802-1829) even
though it was extensively used by Euler (1707-1783) and was used by Leibniz