classify (√3 − ) as rational or irrational, and explain your reasoning.
a) suppose an irrational − rational = rational, then irrational = rational + rational. this means irrational = rational. thus the irrational number
3
minus the rational number
1
3
equals (
3
−
1
3
), which is both rational and irrational.
b) suppose an irrational − rational = rational, then irrational = rational + rational. this means irrational = rational, which is a contradiction. thus the irrational number
3
minus the rational number
1
3
equals the rational number (
3
−
1
3
).
c) suppose an irrational − rational = rational, then irrational = rational + rational. this means irrational = rational, which is a contradiction. thus the irrational number
3
minus the rational number
1
3
equals the irrational number (
3
−
1
3
).
d) suppose an irrational − rational = rational, then irrational = rational + rational. this means irrational = rational, which is a contradiction. thus the irrational number
3
minus the rational number
1
3
equals (
3
−
1
3
), which is neither rational nor irrational.