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
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Third space learning lessons (online with MCQs)
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
- \((\sqrt(a))^2 = a\)
- \(\sqrt(a^2) = a\)
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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
-
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.
-
Reduce the number of square root signs.
By looking for like surds. Perhaps by "factoring".
-
Remove and surds on the denominator of a fraction.
By used of conjugate surds and the difference of squares identity.