Surds - mathematical content knowledge

See also: mathematical-content-knowledge, teaching-mathematics

Numbers left as square (or other) roots that have a decimal that goes on forever. i.e. they cannot be written as a fraction using integers. i.e. they are irrational numbers.

Any number of the form \(\sqrt{a}\), which cannot be written as a fraction of two integers is called a #surd. There is no specific numeric value, at best approximations.

Surds used to be another name for irrational numbers

But now only applied to a root that is irrational

Known surds

• The Golden Ratio is one of the earliest and most famous surds discovered by the Greeks (and perhaps others?)

Related resources

• AMSI module on SURDS TODO work through this

• Third space learning lessons (online with MCQs)

  • Surds
  • Simplifying surds
  • Adding and subtracting surds
  • Multiplying and dividing surds
  • rationale the denominator
  • Rationlising surds

Declarative knowledge

Surds and definitions

• surd
• like surd and unlike surd
• conjugate surd

Basic rules

• For positive numbers, squaring and taking a square root are inverse processes

  • \((\sqrt(a))^2 = a\)
  • \(\sqrt(a^2) = a\)

• Multiplication

  • \(\sqrt(a) \times \sqrt(b) = \sqrt(ab)\)
  • \(a\sqrt(b) \times c\sqrt(d) = ac\sqrt(bd)\)

• Division

  • \(\sqrt(a) \div \sqrt(b) = \sqrt(a/b)\)
  • \(a\sqrt(b) \div c\sqrt(d) = \dfrac{a}{c}\sqrt(b/d)\)

• Addition and subtraction of like surds

  • \(a\sqrt(b) \pm c\sqrt(b) = (a \pm c)\sqrt(b)\)

Procedural knowledge

Simplifying means trying to

1. Reduce the size of the number under the square root sign (radicand).

   By identifying and factors that are perfect squares and taking them out of the square root sign.

2. Reduce the number of square root signs.

   By looking for like surds. Perhaps by "factoring".

3. Remove and surds on the denominator of a fraction.

   By used of conjugate surds and the difference of squares identity.