1.2 Irrational numbers

• Irrational numbers in the set of real numbers
• Simplification of irrational numbers (surds)
• Operations on irrational numbers: addition, subtraction and multiplication
• Rationalisation of denominators, including conjugates; applications

• identify irrational numbers from a set of real numbers
• simplify irrational numbers in different situations
• perform operations on irrational numbers in different situations
• rationalise irrational numbers in different situations
• apply irrational numbers in different situations
• reflect on the use of irrational numbers in mathematical computations

“How are irrational numbers used in daily life?”

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1. Which real numbers are irrational
   40 min · 1 practice tasks · quiz
   Sort a set of real numbers into rational and irrational, and see why the square root of 2 cannot be a fraction.
2. Simplifying surds
   40 min · 1 practice tasks · quiz
   Write a square root as a whole number times a smaller square root by taking out the largest perfect-square factor.
3. Adding and subtracting surds
   40 min · 1 practice tasks · quiz
   Add and subtract only like surds, after simplifying so that the root parts match.
4. Multiplying surds
   40 min · 1 practice tasks · quiz
   Multiply square roots by multiplying the insides, expand a pair of brackets, and recognise a difference of squares.
5. Rationalising a single root
   40 min · 1 practice tasks · quiz
   Remove one square root from a denominator by multiplying the top and bottom by that root, then simplify.
6. Conjugates
   40 min · 1 practice tasks · quiz
   Rationalise a two-term denominator by multiplying by its conjugate, so the bottom becomes a difference of squares.
7. Mixed surd arithmetic
   40 min · 1 practice tasks · quiz
   Simplify, collect, multiply and rationalise in one expression, and choose the order that keeps the exact value.
8. Exact lengths and exact areas
   40 min · 1 practice tasks · quiz
   Give a diagonal or a circular area in exact irrational form, and use a decimal only as a check.