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Recommended reading:

[Ask Sage question 9950: What are the different real numbers in Sage?]https://ask.sagemath.org/question/9950
[http://sagebook.gforge.inria.fr/english.html Book Computational Mathematics with SageMath]

Some ideas:

use RealField(k) for reals with k bits of precision
use Arb for arbitrary precision, via RealBallField or ComplexBallField

Chopping (also know as truncating) might mean different things:

truncate the binary or decimal expansion?
absolute or relative precision, i.e. do you truncate to base^-N or does it depend on the size of the number?

For example, truncating the following numbers to 2 decimal places:

$$ a = 23456.789 $$ $$ b = 2.3456789 $$ $$ c = 0.00023456789 $$

do you expect the result to be

$$ a' = 23456.78 $$ $$ b' = 2.34 $$ $$ c' = 0.0 $$

$$ a' = 23000.00 $$ $$ b' = 2.3 $$ $$ c' = 0.0023 $$

The following function could be a starting point:

def chop(x, precision=3, base=10, absolute=True):
    bp = base^precision
    if absolute:
        return RDF(floor(bp*x)/bp)
    # otherwise truncate to relative precision
    size = floor(log(x, base)) + 1
    bs = base^size
    return chop(x/bs, precision=precision, base=base, absolute=True)*bs

Define a printing and formatting function for illustration:

def chop_list(aa):
    format = "%20s %20s %20s"
    print()
    print(format % ('number', 'chop_absolute', 'chop_relative'))
    for a in aa:
        b = chop(a, precision=2, base=10, absolute=True)
        c = chop(a, precision=2, base=10, absolute=False)
        print("%20s %20s %20s" % (a, b, c))

Check out the results:

sage: aa = [23456.789, 2.3456789, 0.0023456789]
sage: chop_list(aa)

              number        chop_absolute        chop_relative
    23456.7890000000             23456.78              23000.0
    2.34567890000000                 2.34   2.3000000000000003
 0.00234567890000000                  0.0               0.0023

Now for rounding instead of truncating, use round instead of floor.