Practice question · Sort into groups
Each extension of the number system is driven by closure. Sort each expression by the smallest system in which it is guaranteed to have an answer.
Groups: Defined in the integers · Needs the rationals · Needs the reals
- 3 divided by 2
- the square root of 2
- the difference of two integers
- the product of two integers
- 1 divided by 7
- pi
Hints
- Ask what operation the expression uses, then ask which system makes that operation always work.
- Integers are closed under subtraction and multiplication but not division.
Show the answer
Defined in the integers: the difference of two integers, the product of two integers
Needs the rationals: 3 divided by 2, 1 divided by 7
Needs the reals: the square root of 2, pi
Why
Each enlargement exists to make one operation total: negatives make subtraction always work, fractions make division by a nonzero number always work, and the reals supply the roots and limits that fractions cannot reach. Closure is the engine of the whole hierarchy.
Practise Number Systems
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