Human ecology
Discrete mathematics

Representations of rational numbers 1
2... 3...

Rational numbers are numbers which have the form m/n where m and n are integers. Examples: 2/3, 22/7, 1/2, etc...

These numbers can also be expressed as decimals.

2/3 = 0.6666666...
22/7 = 3.142857142857...
1/2 = 0.5

A general decimal representation of a rational number, r, might have the form:

r = ABCD.EFGHI...

Where r = (A*(10^3)) + (B*(10^2)) + (C*(10^1)) + (D*(10^0)) + (E/(10^1)) + (F/(10^2)) + (G/(10^3)) + (H/(10^4)) + (I/(10^5)) + ...

For representations in an arbitary base, b, a general form is:
See below for a pseudo c++ representation of 

this equation

void representation_real_and_rational_numbers();
{
real x;
rational r;
integer a[infinity];
integer b;
integer i;

// representation of a real number

x=0;
for (i=-infinity; i (less than) infinity; i++)
x=x+a[i]*b^i;

// (Unfortunately, the (less than) symbol is also the left angle bracket.
// And the left angle bracket marks the start of html tags)

//
// representation of a rational number
//

// if a real number x is also rational then
// there exists a base b for which
// there is a finite sum.

integer negative_limit;
integer positive_limit;

r=0;
for (i=negative_limit; i (less than) positive_limit+1; i++)
r=r+a[i]*b^i;

}

Perhaps this is a bit longer but is it more understandable?

Conversions from...
Base10 to... Base8
0.0 0.0
0.1 0.063146314...
0.2 0.146314631...
0.3 0.231463146...
0.4 0.314631463...
0.5 0.4
0.6 0.463146314...
0.7 0.546314631...
0.8 0.631463146...
0.9 0.714631463...
Conversions from...
Base8 to... Base10
0.0 0.0
0.1 0.125
0.2 0.25
0.3 0.375
0.4 0.5
0.5 0.675
0.6 0.75
0.7 0.875

Conversions from...
Base10 to... Base3
0.0 0.0
0.1 0.002200220022...
0.2 0.012101210121...
0.3 0.022002200220...
0.4 0.101210121012...
0.5 0.111111111111...
0.6 0.121012101210...
0.7 0.200220022002...
0.8 0.210121012101...
0.9 0.220022002200...
Conversions from...
Base3 to... Base10
0.0 0.0
0.1 0.333333333333...
0.2 0.666666666666...



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Created 30/7/99