1.1A

Scientific Notation / Standard Form

  • Any positive real number can be expressed in scientific notation as: $$a \times 10^k \quad \text{where } 1 \le a < 10 \text{ and } k \in \mathbb{Z}$$
  • Positive Exponents ($k > 0$): Represent large magnitudes (e.g., speed of light $3.00 \times 10^8\text{ m/s}$). Shift the decimal point $k$ places to the right.
  • Negative Exponents ($k < 0$): Represent microscopic measurements (e.g., atom radius $1.2 \times 10^{-10}\text{ m}$). Shift the decimal point $|k|$ places to the left.
  • Calculator Representation: On GDC screens, scientific notation appears as 3.5E8 or 1.4E-4. Examiner Warning: Never write 'E' notation in an official IB written answer. Always write $3.5 \times 10^8$.

Solved Example 1: Operations in Standard Form

Calculate $\frac{(4.8 \times 10^7) \times (2.5 \times 10^{-3})}{1.5 \times 10^2}$, giving your answer in the form $a \times 10^k$ where $1 \le a < 10$.

Step 1: Group coefficients and powers of 10: $$\text{Numerator} = (4.8 \times 2.5) \times 10^{7 + (-3)} = 12.0 \times 10^4$$
Step 2: Divide by the denominator: $$\frac{12.0 \times 10^4}{1.5 \times 10^2} = \left(\frac{12.0}{1.5}\right) \times 10^{4 - 2} = 8.0 \times 10^2$$

Final Answer: $8.0 \times 10^2$ (or $8 \times 10^2$)

1.1B

Significant Figures and IB Rounding Rules

  • The IB 3 Significant Figures Rule: Unless specified otherwise in a question, all final answers in IB Mathematics must be given exactly or rounded correctly to 3 significant figures (3 s.f.).
  • Leading Zeros: Zeros preceding non-zero digits are never significant (e.g. $0.00452$ has 3 s.f.).
  • Trapped Zeros: Zeros between non-zero digits are always significant (e.g. $4005$ has 4 s.f., $2.04$ has 3 s.f.).
  • Trailing Zeros: Trailing zeros after a decimal point count as significant (e.g. $3.50$ has 3 s.f.; $5.000$ has 4 s.f.).
  • Premature Rounding Danger: Never round intermediate calculations. Keep full precision in your GDC memory and only round the final stated result to avoid compounding inaccuracies.
1.1C

Percentage Error Formula

  • When an approximation $v_A$ is used instead of an exact value $v_E$, the percentage error $\varepsilon$ is: $$\varepsilon = \left| \frac{v_A - v_E}{v_E} \right| \times 100\%$$
  • Absolute Value $|\dots|$: Ensures percentage error is always reported as a non-negative value ($\varepsilon \ge 0$).
  • Denominator Rule: The denominator is always the exact/actual theoretical value $v_E$, never the rounded or estimated value $v_A$.

Solved Example 2: Percentage Error

A surveyor estimates the area of a triangular plot to be $v_A = 450\text{ m}^2$. The exact measured area using satellite laser telemetry is $v_E = 438.4\text{ m}^2$. Calculate the percentage error.

Step 1: Apply the IB percentage error formula: $$\varepsilon = \left| \frac{450 - 438.4}{438.4} \right| \times 100\% = \left| \frac{11.6}{438.4} \right| \times 100\%$$
Step 2: Calculate and round to 3 s.f.: $$\varepsilon = 0.02646 \times 100\% = 2.64598\% \approx 2.65\%$$

Final Answer: $2.65\%$

Precision, bounds and estimation

  • For a nonzero signed value, standard form has $1\le|a|<10$ in $a10^k$. Zero has no unique normalised exponent. Round the coefficient and renormalise if it becomes10.
  • A length reported as12.3cm to the nearest0.1cm implies $12.25\le L<12.35$ using the usual positive-number rounding convention. Bounds express measurement uncertainty, not just decimal formatting.
  • For positive dimensions, find product bounds by multiplying lower bounds and upper bounds; for a ratio use lower numerator/upper denominator and vice versa. Percentage error relative to an exact value0 is undefined.
  • Estimate the order of magnitude before calculating. Preserve guard digits and report the units; significant figures and decimal places answer different questions.

10 Practice Exercises

Attempt each problem before opening the solution.

Exercise 1 (Paper 1 — 3 Marks): Write $0.00004825$ in standard form $a \times 10^k$, correct to 3 significant figures.
Round $0.00004825$ to 3 significant figures: the fourth significant digit is 5, rounding 2 up to 3: $0.0000483$ [M1]
Shift decimal point 5 places to the right: $a = 4.83$, $k = -5$ [A1]
Combine into scientific notation: $4.83 \times 10^{-5}$ [A1]

Answer: $4.83 \times 10^{-5}$

Exercise 2 (Paper 1 — 4 Marks): Evaluate $\sqrt{\frac{1.84 \times 10^5}{3.2 \times 10^{-3}}}$.
  1. Calculate the exact value.
  2. State your answer in the form $a \times 10^k$, where $1 \le a < 10, k \in \mathbb{Z}$.
(a) Inside the square root: $\frac{1.84}{3.2} \times 10^{5 - (-3)} = 0.575 \times 10^8 = 57,500,000$ [M1]
Take the square root: $\sqrt{57,500,000} \approx 7582.875$ [A1]
(b) Round to 3 s.f.: $7580$ [M1]
Standard form: $7.58 \times 10^3$ [A1]

Answer: (a) $500\sqrt{230}$ exactly (approximately $7582.88$), (b) $7.58 \times 10^3$

Exercise 3 (Paper 1 — 4 Marks): A student estimates the volume of a sphere of radius $r = 4.5\text{ cm}$ using $\pi \approx 3.14$, getting $v_A$. The exact volume using the calculator value of $\pi$ is $v_E$.
  1. Calculate $v_E$ and $v_A$.
  2. Find the percentage error in the student's estimate.
(a) Volume formula $V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (4.5)^3 = 121.5\pi$.
$v_E = 121.5\pi \approx 381.7035\text{ cm}^3$ [A1]
$v_A = \frac{4}{3}(3.14)(4.5)^3 = 121.5 \times 3.14 = 381.51\text{ cm}^3$ [A1]
(b) $\varepsilon = \left|\frac{381.51 - 381.7035}{381.7035}\right| \times 100\%$ [M1]
$\varepsilon = \frac{0.1935}{381.7035} \times 100\% \approx 0.05069\%$ [A1]

Answer: (a) $v_E = 382\text{ cm}^3, v_A = 381.51\text{ cm}^3$, (b) $0.0507\%$ (or $0.051\%$)

Exercise 4 (Paper 2 — 5 Marks): The distance from Earth to a distant star is $4.22\text{ light years}$. Given that $1\text{ light year} \approx 9.461 \times 10^{12}\text{ km}$:
  1. Find the distance in kilometers, expressing your answer in the form $a \times 10^k$, $1 \le a < 10$.
  2. An astronomer approximates the distance to be $4.0 \times 10^{13}\text{ km}$. Calculate the percentage error in this approximation.
(a) Distance = $4.22 \times (9.461 \times 10^{12}) = 39.92542 \times 10^{12}$ [M1]
Convert to scientific notation: $3.992542 \times 10^{13}\text{ km}$ [A1]
To 3 s.f.: $3.99 \times 10^{13}\text{ km}$ [A1]
(b) $\varepsilon = \left|\frac{4.0 \times 10^{13} - 3.992542 \times 10^{13}}{3.992542 \times 10^{13}}\right| \times 100\%$ [M1]
$\varepsilon = \frac{0.007458}{3.992542} \times 100\% \approx 0.1868\%$ [A1]

Answer: (a) $3.99 \times 10^{13}\text{ km}$, (b) $0.187\%$

Exercise 5 (Paper 1 — 3 Marks): A rectangle has dimensions measured as $l = 14.2\text{ cm}$ and $w = 8.6\text{ cm}$, each correct to 1 decimal place. Find the upper and lower bounds for the perimeter of the rectangle.
Measurement degree of precision is $0.1$, so error margin is $\pm 0.05\text{ cm}$.
Bounds for $l$: $[14.15, 14.25)$. Bounds for $w$: $[8.55, 8.65)$. [M1]
Lower bound perimeter: $2(14.15 + 8.55) = 2(22.70) = 45.4\text{ cm}$ [A1]
Upper bound perimeter: $2(14.25 + 8.65) = 2(22.90) = 45.8\text{ cm}$ [A1]

Answer: Lower bound = $45.4\text{ cm}$, Upper bound = $45.8\text{ cm}$

Exercise 6

Write−0.00072in standard form and explain the coefficient condition.

$-7.2\times10^{-4}$; its coefficient has magnitude7.2. Negative numbers require absolute value in the condition.

Exercise 7

Round $9.996\times10^5$ to3significant figures.

The coefficient rounds to10.0; renormalise to $1.00\times10^6$. The trailing zeros state three significant figures.

Exercise 8

A rectangle has measured lengths12.3cm and4.0cm, each to the nearest0.1cm. Bound its area.

$12.25\le L<12.35$, $3.95\le W<4.05$. Thus $48.3875\le A<50.0175$cm². Multiplying rounded reported values gives49.2cm² but hides uncertainty.

Exercise 9

An estimate is48 and the exact value is50. Find absolute and percentage errors. What changes if the exact value is0?

Absolute error2; percentage error $2/50\times100=4\%$. With exact value0 the percentage formula divides by zero, so report absolute error instead.

Exercise 10

Estimate $(6.1\times10^8)/(2.9\times10^{-3})$ before calculating.

Estimate $2\times10^{11}$. Actual coefficient6.1/2.9≈2.10345, so result $2.10\times10^{11}$to3sf. An answer of order10⁵ indicates an exponent error.

Formulae

Official IB AI SL Topic 1 Formula Reference

Scientific Notation:

$$a \times 10^k, \quad 1 \le a < 10, \quad k \in \mathbb{Z}$$

Percentage Error:

$$\varepsilon = \left| \frac{v_A - v_E}{v_E} \right| \times 100\%$$
Top Examiner Pitfalls

Common Exam Mistakes to Avoid

  • Pitfall 1: Dividing by $v_A$ instead of $v_E$. In percentage error, students frequently put the approximate value in the denominator. The formula explicitly requires dividing by the exact true value $v_E$.
  • Pitfall 2: Forgetting the scientific notation bounds ($1 \le a < 10$). Writing $12.4 \times 10^5$ or $0.78 \times 10^3$ loses the accuracy mark $[A1]$. It must strictly be $1.24 \times 10^6$ and $7.8 \times 10^2$.
  • Pitfall 3: Calculator notation on paper. Writing 2.34E5 on your IB exam paper scores 0 marks for communication. Write $2.34 \times 10^5$.
Topic 1.1 Check

Exam Mini Test (5 Multiple Choice Questions)

Score: 0 / 5

1. Which of the following is written correctly in standard form?

2. How many significant figures does $0.05040$ contain?

3. A student records an experimental mass as $8.4\text{ g}$. The true value is $8.0\text{ g}$. What is the percentage error?

4. Evaluate $(3 \times 10^4) \times (6 \times 10^5)$ in standard form.

5. A length is recorded as $25\text{ cm}$ to the nearest centimeter. What is the lower bound for this length?