<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field>

In grade school you learn about rational numbers, which are expressed as the ratio of two integers (whole numbers). Most numbers, like LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEnJiM5NjA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw==, are irrational. Every irrational number has infinite, non-repeating decimal expansion. Maple can give you many digits of this expansion, as in

evalf(Pi,200);

JCJjdz9RSVwmKltIaVcnZmI1QCZRPnEtVEdddTYiW0ciMyVmYHNKQWVdJjRZJVE0Wm1JI0c4bDNbQCl6MTxATURbLkcnKSoqM2lHMWsieUkjZldcKDQjZTV2JCpScHI+JSlHXXpLUVZFWVFLeiplYEVmVEohJCo+

But of course, 200 or 200 trillion digits are as nothing compared to infinity.

You also learn in grade school that 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. As point of fact,

evalf(Pi-22/7);

JCEoKltrNyEiKg==

This approximation gives about 3 accurate digits, and it can be convenient at times to have the simpler rational form. But where did 22/7 come from, anyway? Is it just a coincidence? Can it be improved?

The key to answering this question is an old idea called continued fractions. It's easiest to start these with a rational number to get the idea. I'll pick one based on my phone number. (Note that I can put more than one command on a line, as long as each ends with a semicolon.)

x:= 3383/831; evalf(x);

IyIlJFEkIiRKKQ==

JCIrKHopKjQyJSEiKg==

It's pretty obvious that I can write x as the sum of an integer (whole number) and a rational remainder less than one. The floor function is handy for rounding down to the nearest integer. (The square brackets below give subscripts on the variable names.)

m[1]:= floor(x); remainder:= x - m[1];

IiIl

IyIjZiIkSik=

Since the remainder is less than one, its reciprocal is greater than one.

y:= 1/remainder; evalf(y);

IyIkSikiI2Y=

JCIrd1haMzkhIik=

So we can write y too as the sum of an integer part and remainder:

m[2]:= floor(y); remainder:= y - m[2];

IiM5

IyIiJiIjZg==

Let's keep cranking these out. (I do this by copy/paste, rather than typing the same stuff over and over.)

m[3]:= floor(1/remainder); remainder:= 1/remainder - m[3];

IiM2

IyIiJSIiJg==

m[4]:= floor(1/remainder); remainder:= 1/remainder - m[4];

IiIi

IyIiIiIiJQ==

Aha! Since the reciprocal of this remainder is exactly the integer 4, we have reached the end of the line--no more remainders.

m[5]:= floor(1/remainder); remainder:= 1/remainder - m[5];

IiIl

IiIh

Obviously the sequence of numbers 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 tells us something about x, but what? Let's go back and do things symbolically. First, we write x as the sum of an integer plus a remainder, which we give its own name.

unassign('x'); x= m[1] + r[1];

L0kieEc2IiwmIiIlIiIiJkkickdGJDYjRidGJw==

Next, we did the same trick again, representing 1/LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn as the sum 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In other words,

r[1]=1/(m[2]+r[2]);

LyZJInJHNiI2IyIiIiokLCYiIzlGJyZGJDYjIiIjRichIiI=

Putting this into the previous result (note that %% goes back to the next to last result),

subs( %, %% );

L0kieEc2IiwmIiIlIiIiKiQsJiIjOUYnJkkickdGJDYjIiIjRichIiJGJw==

And so on:

subs( r[2]=1/(m[3]+r[3]), % );

L0kieEc2IiwmIiIlIiIiKiQsJiIjOUYnKiQsJiIjNkYnJkkickdGJDYjIiIkRichIiJGJ0YyRic=

subs( r[3]=1/(m[4]+r[4]), % );

L0kieEc2IiwmIiIlIiIiKiQsJiIjOUYnKiQsJiIjNkYnKiQsJkYnRicmSSJyR0YkNiNGJkYnISIiRidGM0YnRjNGJw==

subs( r[4]=1/m[5], % );

L0kieEc2IiMiJSRRJCIkSik=

The trail ends here because there are no more remainders, and Maple simplifies the result automatically. Equation (with 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) is called the continued fraction form of the original x.

The continued fraction method also works on irrational numbers, like LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNjBRIi5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSQwZW1GJy8lJ3JzcGFjZUdGTC8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJw==

x:= Pi;

SSNQaUclKnByb3RlY3RlZEc=

m[1]:= floor(x); remainder:= x - m[1];

IiIk

LCZJI1BpRyUqcHJvdGVjdGVkRyIiIiEiJEYl

Even though the remainder is not rational, it is still clearly less than one.

evalf(remainder);

JCIqYUVmVCIhIio=

m[2]:= floor(1/remainder); remainder:= 1/remainder - m[2];

IiIo

LCYqJCwmSSNQaUclKnByb3RlY3RlZEciIiIhIiRGJyEiIkYnISIoRic=

m[3]:= floor(1/remainder); remainder:= 1/remainder - m[3];

IiM6

LCYqJCwmKiQsJkkjUGlHJSpwcm90ZWN0ZWRHIiIiISIkRikhIiJGKSEiKEYpRitGKSEjOkYp

m[4]:= floor(1/remainder); remainder:= 1/remainder - m[4];

IiIi

LCYqJCwmKiQsJiokLCZJI1BpRyUqcHJvdGVjdGVkRyIiIiEiJEYrISIiRishIihGK0YtRishIzpGK0YtRitGLUYr

and so on. You can see the continued fraction form emerging in the remainder.

The difference this time is that the process can never stop. If we ever got a zero remainder, then we would be able to write \317\200 as a rational number exactly by simplifying the continued fraction. However, by terminating the process prematurely (tossing out the remainder as if it were zero) we do get rational approximations to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNjBRIi5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSQwZW1GJy8lJ3JzcGFjZUdGTC8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJw==

x[1]:= m[1]+r[1]; subs(r[1]=0,x[1]);

LCYiIiQiIiImSSJyRzYiNiNGJEYk

IiIk

x[2]:= subs( r[1]=1/(m[2]+r[2]), x[1] ); subs(r[2]=0,x[2]);

LCYiIiQiIiIqJCwmIiIoRiQmSSJyRzYiNiMiIiNGJCEiIkYk

IyIjQSIiKA==

x[3]:= subs( r[2]=1/(m[3]+r[3]), x[2] ); subs(r[3]=0,x[3]);

LCYiIiQiIiIqJCwmIiIoRiQqJCwmIiM6RiQmSSJyRzYiNiNGI0YkISIiRiRGL0Yk

IyIkTCQiJDEi

x[4]:= subs( r[3]=1/(m[4]+r[4]), x[3] ); subs(r[4]=0,x[4]);

LCYiIiQiIiIqJCwmIiIoRiQqJCwmIiM6RiQqJCwmRiRGJCZJInJHNiI2IyIiJUYkISIiRiRGMkYkRjJGJA==

IyIkYiQiJDgi

evalf( [3,22/7,333/106,355/113,Pi] );

NyckIiIkIiIhJCIrVnImRzkkISIqJCIrTSU0OjkkRigkIis/SGZUSkYoJCIrYUVmVEpGKA==

As you can see, the last approximation gets seven correct significant digits.