Uvero — STEM Learning

Real Number Types and Irrationality

Natural, integer, rational and irrational numbers nest within the real numbers. A terminating or recurring decimal is rational; a non-terminating non-recurring decimal is irrational. At Grade 10, fluent symbolic work must stay connected to graphs, tables, diagrams and real quantities.

Worked reasoning

$\sqrt2$ cannot be written as a ratio of integers.
The other choices are rational.

Which number is irrational?

$\sqrt2$

$0.125$

$7/9$

$-4$

Which statement best captures the central mathematical idea in Real Number Types and Irrationality?

A terminating or recurring decimal is rational; a non-terminating non-recurring decimal is irrational.

A memorised rule may be used without checking signs, domains, units, graph family or geometric hypotheses.

A memorised formula is valid even when its conditions are not met.

The diagram or notation replaces the need to justify each mathematical step.

When starting a problem about Real Number Types and Irrationality, which move is most reliable?

Identify the mathematical family, list the given quantities and state the domain or geometric conditions for real number types and irrationality.

Round every value before identifying the mathematical structure.

Choose the longest available formula and substitute immediately.

Assume the desired conclusion and work backwards without checking equivalence.

Which statement is a misconception that must be rejected when working with Real Number Types and Irrationality?

A memorised rule may be used without checking signs, domains, units, graph family or geometric hypotheses.

A terminating or recurring decimal is rational; a non-terminating non-recurring decimal is irrational.

Identify the mathematical family, list the given quantities and state the domain or geometric conditions for real number types and irrationality.

Use substitution, a second representation or a magnitude estimate to verify the result.