Scientific Notation (Standard Form $a \times 10^k$)
- Formal Definition: A number is written in scientific notation (standard form) when expressed as:
$$a \times 10^k \quad \text{where } 1 \le |a| < 10 \text{ and } k \in \mathbb{Z}$$
• If the original number is $\ge 10$, then $k > 0$ (e.g., $450\,000 = 4.5 \times 10^5$).
• If the original number is between $0$ and $1$, then $k < 0$ (e.g., $0.00072 = 7.2 \times 10^{-4}$). - Computations in Standard Form:
• Multiplication: $(a \times 10^p) \times (b \times 10^q) = (ab) \times 10^{p+q}$. Adjust $ab$ if $ab \ge 10$.
• Division: $\frac{a \times 10^p}{b \times 10^q} = \left(\frac{a}{b}\right) \times 10^{p-q}$. Adjust $\frac{a}{b}$ if $\frac{a}{b} < 1$.
• Addition/Subtraction: Convert both numbers to share the same power of $10$ before combining coefficients: $$(3.2 \times 10^4) + (4.5 \times 10^3) = (3.2 \times 10^4) + (0.45 \times 10^4) = (3.2 + 0.45) \times 10^4 = 3.65 \times 10^4$$
Example 1 · Criterion A & D
Scientific Notation Multiplication and Division
The speed of light in vacuum is approximately $c = 3.0 \times 10^8\text{ m s}^{-1}$. The mean distance from the Sun to Mars is $2.28 \times 10^{11}\text{ m}$.
(a) Calculate the time (in seconds) for light from the Sun to reach Mars. Give your answer in scientific notation.
(b) Convert this time into minutes to 3 significant figures.
- (a) Time calculation: $$t = \frac{\text{Distance}}{c} = \frac{2.28 \times 10^{11}}{3.0 \times 10^8} = \left(\frac{2.28}{3.0}\right) \times 10^{11 - 8} = 0.76 \times 10^3\text{ s}$$ Normalizing into standard form ($1 \le a < 10$): $$t = (7.6 \times 10^{-1}) \times 10^3 = 7.6 \times 10^2\text{ seconds}$$
- (b) Conversion to minutes: $$t = \frac{760\text{ s}}{60\text{ s min}^{-1}} = 12.666\dots\text{ minutes} \approx 12.7\text{ minutes (to 3 s.f.)}$$
Answers: (a) $7.6 \times 10^2\text{ s}$; (b) $12.7\text{ minutes}$.
Significant Figures, Rounding & Upper/Lower Error Bounds
- Rules for Counting Significant Figures (s.f.):
• All non-zero digits are significant ($4.27 \to 3$ s.f.).
• Captive zeros between non-zero digits are significant ($5004 \to 4$ s.f., $3.08 \to 3$ s.f.).
• Leading zeros before the first non-zero digit are never significant ($0.0075 \to 2$ s.f.).
• Trailing zeros in a decimal portion are significant ($4.500 \to 4$ s.f.). - Degree of Accuracy (IB Standard): Unless stated otherwise, answers must be given to 3 significant figures (or exact values).
- Upper and Lower Error Bounds: When a value $x$ is rounded to the nearest unit of measurement $U$: $$\text{Maximum Absolute Error} = \frac{U}{2}$$ $$\text{Lower Bound (LB)} = x - \frac{U}{2}, \quad \text{Upper Bound (UB)} = x + \frac{U}{2}$$ $$\text{Error Interval: } \text{LB} \le x < \text{UB}$$
- Bounds for Operations:
• Sum: $\text{UB}(A + B) = \text{UB}_A + \text{UB}_B$, $\text{LB}(A + B) = \text{LB}_A + \text{LB}_B$
• Difference: $\text{UB}(A - B) = \text{UB}_A - \text{LB}_B$, $\text{LB}(A - B) = \text{LB}_A - \text{UB}_B$
• Product: $\text{UB}(A \times B) = \text{UB}_A \times \text{UB}_B$, $\text{LB}(A \times B) = \text{LB}_A \times \text{LB}_B$ (for positive quantities)
• Quotient: $\text{UB}\left(\frac{A}{B}\right) = \frac{\text{UB}_A}{\text{LB}_B}$, $\text{LB}\left(\frac{A}{B}\right) = \frac{\text{LB}_A}{\text{UB}_B}$
Example 2 · Criterion A & D
Calculating Error Bounds for Area and Speed
A rectangle has length $L = 8.4\text{ cm}$ and width $W = 5.6\text{ cm}$, each rounded to $1\text{ decimal place}$.
(a) State the lower and upper bounds of $L$ and $W$.
(b) Calculate the lower bound and upper bound of the area $A$.
(c) State the area value to an appropriate degree of accuracy.
- (a) Individual bounds (unit $U = 0.1$, error margin $= 0.05$): $$8.35 \le L < 8.45\text{ cm}$$ $$5.55 \le W < 5.65\text{ cm}$$
- (b) Area bounds: $$\text{LB}(A) = \text{LB}(L) \times \text{LB}(W) = 8.35 \times 5.55 = 46.3425\text{ cm}^2$$ $$\text{UB}(A) = \text{UB}(L) \times \text{UB}(W) = 8.45 \times 5.65 = 47.7425\text{ cm}^2$$
- (c) Degree of accuracy: Both bounds round to $47\text{ cm}^2$ (to 2 s.f.) or $50\text{ cm}^2$ (to 1 s.f.). Because the tens digit is $4$ and units digit differs ($46$ vs $47$), $A = 47\text{ cm}^2$ is the most reliable representation.
Answers: (a) $L \in [8.35, 8.45)$, $W \in [5.55, 5.65)$; (b) $\text{LB} = 46.3425\text{ cm}^2$, $\text{UB} = 47.7425\text{ cm}^2$; (c) $47\text{ cm}^2$.
Surds: Simplification, Arithmetic & Conjugates
- Fundamental Radical Laws: $$\sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \quad (a, b \ge 0)$$ $$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \quad (a \ge 0, b > 0)$$
- Simplifying to $k\sqrt{m}$: Factor out the largest square number ($4, 9, 16, 25, 36, 49, \dots$): $$\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}$$
- Adding/Subtracting Like Surds: Surds with identical radicands combine algebraically: $$4\sqrt{18} - 2\sqrt{50} = 4(3\sqrt{2}) - 2(5\sqrt{2}) = 12\sqrt{2} - 10\sqrt{2} = 2\sqrt{2}$$
- Expanding Brackets: $$(3 + 2\sqrt{5})(4 - \sqrt{5}) = 12 - 3\sqrt{5} + 8\sqrt{5} - 2(5) = 2 + 5\sqrt{5}$$
- Conjugate Difference of Squares: $$(a + \sqrt{b})(a - \sqrt{b}) = a^2 - (\sqrt{b})^2 = a^2 - b \in \mathbb{Q}$$
Example 3 · Criterion A & C
Rationalising Monomial and Binomial Denominators
Rationalise the denominator and simplify each expression:
(a) $\frac{12}{\sqrt{6}}$ (b) $\frac{4}{3 - \sqrt{5}}$ (c) $\frac{\sqrt{3} + 1}{\sqrt{3} - 1}$
- (a) Monomial denominator: Multiply numerator and denominator by $\sqrt{6}$: $$\frac{12}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{12\sqrt{6}}{6} = 2\sqrt{6}$$
- (b) Binomial denominator: Multiply by the conjugate $3 + \sqrt{5}$: $$\frac{4}{3 - \sqrt{5}} \times \frac{3 + \sqrt{5}}{3 + \sqrt{5}} = \frac{4(3 + \sqrt{5})}{3^2 - (\sqrt{5})^2} = \frac{4(3 + \sqrt{5})}{9 - 5} = \frac{4(3 + \sqrt{5})}{4} = 3 + \sqrt{5}$$
- (c) Conjugate with square: $$\frac{\sqrt{3} + 1}{\sqrt{3} - 1} \times \frac{\sqrt{3} + 1}{\sqrt{3} + 1} = \frac{(\sqrt{3} + 1)^2}{(\sqrt{3})^2 - 1^2} = \frac{3 + 2\sqrt{3} + 1}{3 - 1} = \frac{4 + 2\sqrt{3}}{2} = 2 + \sqrt{3}$$
Answers: (a) $2\sqrt{6}$; (b) $3 + \sqrt{5}$; (c) $2 + \sqrt{3}$.
Express each of the following in scientific notation ($a \times 10^k$):
(a) $48\,500\,000$ (b) $0.0000307$ (c) $650 \times 10^4$
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(a) $48\,500\,000 = 4.85 \times 10^7$ [A1]
(b) $0.0000307 = 3.07 \times 10^{-5}$ [A1]
(c) $650 \times 10^4 = (6.5 \times 10^2) \times 10^4 = 6.5 \times 10^6$ [A1]
Evaluate each expression without a calculator, giving your answer in standard form:
(a) $(4.2 \times 10^5) \times (3.0 \times 10^{-2})$ [1 mark]
(b) $\frac{8.4 \times 10^7}{2.1 \times 10^3}$ [1 mark]
(c) $(7.5 \times 10^4) + (2.5 \times 10^3)$ [1 mark]
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(a) Multiplication: [1 mark]
$(4.2 \times 3.0) \times 10^{5 + (-2)} = 12.6 \times 10^3 = 1.26 \times 10^4$ [A1]
(b) Division: [1 mark]
$\left(\frac{8.4}{2.1}\right) \times 10^{7 - 3} = 4.0 \times 10^4$ [A1]
(c) Addition: [1 mark]
$(7.5 \times 10^4) + (0.25 \times 10^4) = (7.5 + 0.25) \times 10^4 = 7.75 \times 10^4$ [A1]
State the number of significant figures in each value:
(a) $0.004050\text{ kg}$ (b) $800.0\text{ m}$ (c) $5.20 \times 10^6\text{ km}$
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(a) $0.004050$: [1 mark] Leading zeros do not count; $4, 0, 5, 0$ count $\implies \mathbf{4\text{ s.f.}}$ [A1]
(b) $800.0$: [1 mark] The explicit decimal makes all zeros significant $\implies \mathbf{4\text{ s.f.}}$ [A1]
(c) $5.20 \times 10^6$: [1 mark] In scientific notation, all digits in coefficient are significant $\implies \mathbf{3\text{ s.f.}}$ [A1]
Simplify each radical expression to the form $k\sqrt{m}$ where $m$ has no square factors:
(a) $\sqrt{108}$ (b) $\sqrt{200} - \sqrt{32} + \sqrt{50}$
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(a) $\sqrt{108}$: [2 marks]
$\sqrt{108} = \sqrt{36 \times 3} = \sqrt{36} \times \sqrt{3} = 6\sqrt{3}$ [A2]
(b) Combining surds: [2 marks]
$\sqrt{200} = \sqrt{100 \times 2} = 10\sqrt{2}$ [M1]
$\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$
$\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$
Expression $= 10\sqrt{2} - 4\sqrt{2} + 5\sqrt{2} = 11\sqrt{2}$ [A1]
Rationalise the denominator of each fraction, expressing your answer in simplest form:
(a) $\frac{15}{2\sqrt{5}}$ [2 marks]
(b) $\frac{6}{4 - \sqrt{10}}$ [2 marks]
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(a) Monomial: [2 marks]
$\frac{15}{2\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{15\sqrt{5}}{2 \times 5} = \frac{15\sqrt{5}}{10} = \frac{3\sqrt{5}}{2}$ [A2]
(b) Binomial: [2 marks]
Multiply by conjugate $4 + \sqrt{10}$:
$\frac{6(4 + \sqrt{10})}{4^2 - (\sqrt{10})^2} = \frac{6(4 + \sqrt{10})}{16 - 10} = \frac{6(4 + \sqrt{10})}{6} = 4 + \sqrt{10}$ [A2]
Expand and simplify each product of radical binomials:
(a) $(2\sqrt{3} + 5)(2\sqrt{3} - 5)$ [2 marks]
(b) $(3\sqrt{2} - 4)^2$ [2 marks]
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(a) Difference of squares: [2 marks]
$(2\sqrt{3})^2 - 5^2 = 4(3) - 25 = 12 - 25 = -13$ [A2]
(b) Binomial expansion: [2 marks]
$(3\sqrt{2})^2 - 2(3\sqrt{2})(4) + 4^2 = 9(2) - 24\sqrt{2} + 16 = 18 - 24\sqrt{2} + 16 = 34 - 24\sqrt{2}$ [A2]
Given that $\frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}} = a + b\sqrt{21}$, where $a, b \in \mathbb{Q}$:
Find the exact values of $a$ and $b$. Show all steps clearly.
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Step 1: Multiply by conjugate of denominator [1 mark]
Multiply numerator and denominator by $\sqrt{7} + \sqrt{3}$ [M1]:
$$\frac{(\sqrt{7} + \sqrt{3})(\sqrt{7} + \sqrt{3})}{(\sqrt{7} - \sqrt{3})(\sqrt{7} + \sqrt{3})}$$
Step 2: Expand numerator and denominator [2 marks]
Denominator: $(\sqrt{7})^2 - (\sqrt{3})^2 = 7 - 3 = 4$ [A1]
Numerator: $(\sqrt{7})^2 + 2\sqrt{7}\sqrt{3} + (\sqrt{3})^2 = 7 + 2\sqrt{21} + 3 = 10 + 2\sqrt{21}$ [A1]
Step 3: Simplify and extract coefficients [1 mark]
$$\frac{10 + 2\sqrt{21}}{4} = \frac{10}{4} + \frac{2}{4}\sqrt{21} = \frac{5}{2} + \frac{1}{2}\sqrt{21}$$
$$a = \frac{5}{2}, \quad b = \frac{1}{2}$$ [A1]
A sprinter completes a $100\text{ m}$ dash.
• The distance $d = 100\text{ m}$ is measured correct to the nearest metre.
• The time $t = 9.8\text{ seconds}$ is recorded correct to the nearest $0.1\text{ second}$.
(a) Write down the lower and upper bounds of $d$ and $t$. [2 marks]
(b) Calculate the maximum possible average speed of the sprinter in $\text{m s}^{-1}$. Give your answer to 3 significant figures. [2 marks]
(c) Calculate the minimum possible average speed to 3 significant figures. [1 mark]
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(a) Bounds of $d$ and $t$: [2 marks]
$d$: unit $1\text{ m} \implies 99.5 \le d < 100.5\text{ m}$ [A1]
$t$: unit $0.1\text{ s} \implies 9.75 \le t < 9.85\text{ s}$ [A1]
(b) Maximum average speed: [2 marks]
To maximize $\text{Speed} = \frac{d}{t}$, use $\text{UB}(d)$ and $\text{LB}(t)$ [M1]:
$$\text{Speed}_{\max} = \frac{100.5}{9.75} \approx 10.3077\text{ m s}^{-1} \approx \mathbf{10.3\text{ m s}^{-1} \text{ (3 s.f.)}}$$ [A1]
(c) Minimum average speed: [1 mark]
$$\text{Speed}_{\min} = \frac{\text{LB}(d)}{\text{UB}(t)} = \frac{99.5}{9.85} \approx 10.1015\text{ m s}^{-1} \approx \mathbf{10.1\text{ m s}^{-1} \text{ (3 s.f.)}}$$ [A1]
The mass of planet Earth is approximately $M = 5.97 \times 10^{24}\text{ kg}$ and the mass of the Moon is $m = 7.35 \times 10^{22}\text{ kg}$.
(a) Calculate how many times more massive Earth is than the Moon. Give your answer to 3 significant figures. [2 marks]
(b) Calculate the combined mass of the Earth-Moon system in scientific notation ($a \times 10^k$). [2 marks]
(c) Express the difference between their masses in scientific notation. [1 mark]
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(a) Ratio: [2 marks]
$\text{Ratio} = \frac{5.97 \times 10^{24}}{7.35 \times 10^{22}} = \left(\frac{5.97}{7.35}\right) \times 10^2 \approx 0.81224 \times 10^2 \approx \mathbf{81.2\text{ times}}$ [A2]
(b) Combined mass: [2 marks]
Convert to same power: $7.35 \times 10^{22} = 0.0735 \times 10^{24}$ [M1]
$M + m = (5.97 + 0.0735) \times 10^{24} = \mathbf{6.04 \times 10^{24}\text{ kg}}$ (to 3 s.f.) [A1]
(c) Difference: [1 mark]
$M - m = (5.97 - 0.0735) \times 10^{24} = \mathbf{5.90 \times 10^{24}\text{ kg}}$ (to 3 s.f.) [A1]
Consider the expression $P_n = (\sqrt{n+1} - \sqrt{n})(\sqrt{n+1} + \sqrt{n})$ for $n \in \mathbb{Z}^+$.
(a) Simplify $P_n$ algebraically for any positive integer $n$. [1 mark]
(b) Hence, show that $\frac{1}{\sqrt{n+1} + \sqrt{n}} = \sqrt{n+1} - \sqrt{n}$. [1 mark]
(c) Use this identity to evaluate the telescoping sum without a calculator:
$$S = \frac{1}{\sqrt{2} + 1} + \frac{1}{\sqrt{3} + \sqrt{2}} + \frac{1}{\sqrt{4} + \sqrt{3}} + \dots + \frac{1}{\sqrt{100} + \sqrt{99}}$$
Show all intermediate terms and state your final exact value. [3 marks]
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(a) Difference of squares: [1 mark]
$P_n = (\sqrt{n+1})^2 - (\sqrt{n})^2 = (n + 1) - n = 1$ [A1]
(b) Conjugate reciprocal: [1 mark]
Divide both sides by $(\sqrt{n+1} + \sqrt{n})$:
$\sqrt{n+1} - \sqrt{n} = \frac{1}{\sqrt{n+1} + \sqrt{n}}$ $\blacksquare$ [R1]
(c) Telescoping sum evaluation: [3 marks]
Rewrite each term using the identity [M1]:
$T_1 = \sqrt{2} - 1$
$T_2 = \sqrt{3} - \sqrt{2}$
$T_3 = \sqrt{4} - \sqrt{3}$
$\dots$
$T_{99} = \sqrt{100} - \sqrt{99}$
Notice that all intermediate terms cancel in pairs [M1]:
$S = -1 + (\sqrt{2} - \sqrt{2}) + (\sqrt{3} - \sqrt{3}) + \dots + (\sqrt{99} - \sqrt{99}) + \sqrt{100}$
$S = -1 + \sqrt{100} = -1 + 10 = \mathbf{9}$ [A1]
Forms of Number Quick Reference
Scientific Notation / Standard Form
Upper & Lower Error Bounds
Surd Laws & Arithmetic
Conjugate Rationalisation
Top 5 Common MYP Pitfalls
1. Violating the Range of $a$ in $a \times 10^k$
Writing $12.5 \times 10^4$ or $0.75 \times 10^{-3}$ loses Criterion C communication marks. The leading coefficient must satisfy $1 \le |a| < 10$: write $1.25 \times 10^5$ and $7.5 \times 10^{-4}$.
2. Adding Radicands Under Square Roots
$\sqrt{a + b} \ne \sqrt{a} + \sqrt{b}$! For example, $\sqrt{9 + 16} = \sqrt{25} = 5$, whereas $\sqrt{9} + \sqrt{16} = 3 + 4 = 7$. Square roots do not distribute across addition.
3. Mismatching Bounds in Division & Subtraction
To find the maximum of $\frac{A}{B}$, students often compute $\frac{\text{UB}_A}{\text{UB}_B}$. This is wrong! Dividing by a larger number reduces the result. The true maximum is always $\frac{\text{UB}_A}{\text{LB}_B}$.
4. Dropping Signs in Conjugate Products
In $(2 - \sqrt{7})(2 + \sqrt{7}) = 2^2 - (\sqrt{7})^2 = 4 - 7 = -3$. Do not accidentally write $4 + 7 = 11$. Remember that $(A - B)(A + B) = A^2 - B^2$.
Forms of Number Knowledge Check
Answer these 5 questions to test standard form, bounds, and surd simplification skills.
Which of the following represents $(5.0 \times 10^7) \times (6.0 \times 10^{-3})$ correctly in standard form?
What is the simplest radical form of $\sqrt{75} + \sqrt{48}$?
A plank has a length of $3.6\text{ m}$ rounded to the nearest $0.1\text{ m}$. What is the lower bound of its length?
What is the result of rationalising $\frac{10}{\sqrt{5} + 1}$?
A car travels distance $D = 120\text{ km}$ (to nearest $10\text{ km}$) in time $T = 2\text{ hours}$ (to nearest hour). What is the upper bound of the average speed?