Number Systems, Surds and Logarithms
This topic extends work on rational and irrational numbers and develops exact methods for manipulating surds, powers, standard form and logarithms.
Learning objectives
After studying these notes, learners should be able to define the main terms, apply the central rules and formulas, solve structured examples and explain each answer clearly.
Key ideas
- Natural, integer, rational, irrational and real numbers
- Recurring decimals and fractions
- Scientific notation and significant figures
- Simplifying and rationalising surds
- Index and logarithm laws
Essential rules and formulas
- sqrt(ab) = sqrt(a)sqrt(b)
- (a+b)(a-b) = a^2-b^2
- log base a of x = y means a^y = x
- log(x/y) = log x – log y
Worked example
Rationalise 3/(2+sqrt(5)) by multiplying numerator and denominator by 2-sqrt(5), giving 3(sqrt(5)-2).
Study method
Begin every problem by listing the information supplied and identifying the rule that connects it to the unknown quantity. Keep algebraic steps on separate lines, include units where required and check that the final result is reasonable.
Practice questions
- Write 0.000472 in standard form.
- Simplify sqrt(72) – 2sqrt(8).
- Solve 3^(x+1) = 81.
Revision checklist
- Can you state each definition without looking at the notes?
- Can you select the correct formula and substitute values accurately?
- Can you explain the reasoning behind each step?
- Can you solve a similar examination question independently?
