Curriculum placement
Algebra 2. This describes the skill, not a learner estimate.
Which number systems stay closed
Justify rational and irrational sums and products including nonzero restrictions
Problem types
- Rational and irrational sums and products
Builds on
- Telling rationals from irrationals
- Positive and negative numbers · Earlier idea
Sorting numbers into whole, integer and rational sets begins checking whether each set is closed under an operation.
Leads to
- Calculating with complex numbers · Extend the idea
Real numbers are not closed under the square root, and the complex numbers extend them so that every number has one.
- The quadratic formula · See it applied
A root such as (−b + √d) / 2a is a rational number plus a nonzero rational multiple of √d. When √d is irrational, the closure rules make the whole root irrational.
Related skills
- Which matrix rules hold · Related idea
Closure and properties are checked for matrices as for numbers: matrices add and multiply, but multiplication is not commutative.
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