IB Mathematics AI SL Operations with numbers Study Notes - New Syllabus

ROUNDING – SCIENTIFIC FORM – % ERROR

♦ Rounding of Numbers

The numerical answer to a problem is not always exact, and we often need to round numbers.

Decimal Places (d.p.) vs. Significant Figures (s.f.)
Consider the number
$ 123.4567 $

♦ Rounding to decimal places:
$\text{1 d.p.: 123.5}$
$\text{2 d.p.: 123.46}$
$\text{3 d.p.: 123.457}$

♦ Rounding to significant figures:
$\text{4 s.f.: 123.5}$
$\text{5 s.f.: 123.46}$
$\text{ 6 s.f.: 123.457}$
$\text{2 s.f.: 120}$
$\text{1 s.f.: 100}$

♦ Rule for rounding
If the next digit is 0-4, the digit remains unchanged.
If the next digit is 5-9, the digit increases by 1.

♦ Scientific Notation $(a × 10^k)$

Any number can be written as:
$ a \times 10^k \quad \text{where} \quad 1 \leq a < 10 $

$ 123.4567 = 1.234567 \times 10^2 $
(Moved the decimal point 2 places to the left ⇒ k = 2)

For small numbers:
$ 0.000012345 = 1.2345 \times 10^{-5} $
(Moved the decimal point 5 places to the right ⇒ k = -5)

Scientific form to 3 s.f.:

$ 1.2345 \times 10^2 \equiv 1.23 \times 10^2 $
Calculator notation:
\( 1.2345E+02 \) means \( 1.2345 \times 10^2 \)

Example 
Give the standard form of:
$ s = 4.501 \times 10^7 \quad t = 4.501 \times 10^{-7} $

♦ Percentage Error

When approximating a value (e.g., π), we introduce an error.

For π ≈ 3.14 (to 3 s.f.):

Absolute error: \( |π – 3.14| = 0.00159265… \)

Percentage error:
$ \epsilon = \left| \frac{V_A – V_E}{V_E} \right| \times 100\% $
where \( V_E \) is the exact value and \( V_A \) is the approximate value.