By Mathematical definition: a rational number is an integer divided by
an integer.
An irrational number is a number that can not be exactly represented
as a ratio of two integers.
let a, b, c, d be integres: 1, 2, 3, 4, 5, 6, ...
then a/b is a rational number, and c/d is a rational number
an integer plus an integer is an integer.
an integer times an integer is an integer.
all integers are rational numbers divided by 1
a rational number plus a rational number is a rational number
a/b + c/d = ad/bd + cb/db = (ad + cb)/bd a rational number
a rational number minus a rational number is a rational number
a/b - c/d = ad/bd - cb/db = (ad - cb)/bd a rational number
(we may want to require ad > cb for a positive integer)
a rational number times a rational number is a rational number
(a/b) * (c/d) = ac/bd a rational number
a rational number divided by a rational number is a rational number
(a/b) / (c/d) = ad/bc a rational number
the reciprocal of a rational number is a rational number
1/(a/b) = b/a a rational number
the rational number 1/3 written as a decimal fraction is
1/3 = 0.33333333333333333333333333333333333333333333333333333 ...
the rational number 1/10 written as a DECIMAL fraction is
1/10 = 0.1 exactly
the rational number 1/10 written as a binary fraction is
1/10 = 0.0001100110011001100110011001100110011001100110011 ...
= 1/16 + 1/32 + 1/256 + 1/512 + ...
taylor(sqrt(1+x),x,0,9);
2 3 4 5 6 7 8 9
X X X 5 X 7 X 21 X 33 X 429 X 715 X
sqrt(1+x) = 1 + - - -- + -- - ---- + ---- - ----- + ----- - ------ + ------ + . . .
2 8 16 128 256 1024 2048 32768 65536
sqrt(1+0) = sqrt(1) = 1
sqrt(1+3) = sqrt(4) = 1 + 3/2 - 9/8 + 27/16 - 5*81/128 + ... = 2
sqrt(1+1) = sqrt(2) = 1 + 1/2 - 1/8 + 1/16 - 5/128 + ... = 1.414213562373 ...
taylor(exp(x),x,0,7);
2 3 4 5 6 7
X X X X X X
1 + X + -- + -- + -- + --- + --- + ---- + . . .
2 6 24 120 720 5040
taylor(atan(x),x,0,9);
3 5 7 9
X X X X
X - -- + -- - -- + -- + . . .
3 5 7 9
taylor(atan(x+1),x,0,9);
2 3 5 6 7 9
%PI X X X X X X X
--- + - - -- + -- - -- + -- - --- + --- + . . .
4 2 4 12 40 48 112 288
Last updated 7/3/18