11.1 Numbers and Applications – Comprehensive Notes (Grade 11 Advanced Mathematics M1)

11.1.1 BASIC NUMERACY

Purpose: Develop familiarity with real numbers and their properties; groundwork for real-world measurement, estimation, and algebraic manipulation.
Real numbers and the real number system
- Real numbers: all numbers on the number line, denoting a quantity along a continuous line: $\mathbb{R}$ .
- Subsets of real numbers:
- Natural numbers: counting numbers starting from 1: $\mathbb{N} = {1,2,3,\dots}$
- Whole numbers: natural numbers with 0: $W = {0,1,2,3,\dots}$
- Integers: all positive, negative and zero: $\mathbb{Z}={\dots,-2,-1,0,1,2,\dots}$
- Rational numbers: numbers expressible as a ratio of two integers, Q=\left{\frac{a}{b} : a,b\in\mathbb{Z},\; b\neq 0\right}
- Irrational numbers: numbers not expressible as a ratio of integers (e.g., $\sqrt{2}, \pi$ ).
- Real numbers include both rational and irrational numbers; collectively form $\mathbb{R} = \mathbb{Q} \cup \mathbb{I}$ .
Fractions, decimals, and their types
- Fractions represent parts of a whole; decimal representations may be terminating (finite) or recurring (repeating).
- Proper vs. improper fractions; mixed numbers vs. improper fractions.
Plotting real numbers and number line concepts
- Integers on a number line; opposites defined by equal distance from zero in opposite directions.
- Absolute value: distance from zero; $|n|$ is the distance of the number $n$ from 0.
- Plotting examples: locate numbers like $-5, -1, 0, 1, 5$ on a number line; distance between numbers is measured by absolute value of their difference.
Properties of Real Numbers (axioms; basic algebraic rules for addition and multiplication)
- Axiom 1: Closure property of addition: if $a,b\in\mathbb{R}$ then $a+b\in\mathbb{R}$ .
- Axiom 2: Closure property of multiplication: if $a,b\in\mathbb{R}$ then $ab\in\mathbb{R}$ .
- Axiom 3: Commutative property of addition: $a+b=b+a$ .
- Axiom 4: Commutative property of multiplication: $ab=ba$ .
- Axiom 5: Associative property of addition: $(a+b)+c = a+(b+c)$ .
- Axiom 6: Associative property of multiplication: $(ab)c = a(bc)$ .
- Axiom 7: Identity property of addition: $a+0 = a$ .
- Axiom 8: Identity property of multiplication: $a\cdot 1 = a$ .
- Axiom 9: Inverse property of addition: for every $a$ there exists $-a$ such that $a+(-a)=0$ .
- Axiom 10: Inverse property of multiplication: for every nonzero $a$ there exists $a^{-1}$ such that $a\cdot a^{-1}=1$ .
- Axiom 11: Zero property of multiplication: $a\cdot 0 = 0\; (\forall a\in\mathbb{R})$ .
- Axiom 12: Distributive property of multiplication over addition: $a(b+c)=ab+ac$ .
Surds (radicals)
- Surds are irrational numerical expressions involving roots; e.g., $\sqrt{2}\,$ , $\sqrt{3}$ , etc.
- Index notation and fractional exponents relate surds to powers: e.g., $a^{m/n}=\sqrt[n]{a^{m}}$ (interpreting the radicand and index).
- Key Surd Laws (conceptual, as used in the text):
- Law 1 (surd factorization): factors of a radicand can be separated into surds: e.g., $\sqrt{ab}=\sqrt{a}\sqrt{b}$ when appropriate.
- Law 2 (numerator/denominator factorization for fractions under roots): separate surds in a fraction under a radical when possible.
- Law 3 (power of a surd): if a surd is raised to a power, distribute the exponent to the radicand: $\left(\sqrt{a}\right)^n = a^{n/2}$ (and analogs for generalized fractional powers).
- Law 4 (conjugates and difference of squares): the product of conjugates gives a difference of squares, e.g., $(a+b)(a-b) = a^2 - b^2$ .
- Operations with surds:
- Similar surds can be added/subtracted; unlike surds require simplification via laws.
- Multiplication/division use surd laws; rationalizing denominators is common.
Recurring and non-recurring decimals
- Recurring decimals: digits repeat forever after the decimal point; notation often uses a dot/line over repeating digits.
- Non-recurring decimals (terminating) have a finite number of digits after the decimal.
Significant figures (sf)
- sf are the digits that carry meaning contributing to precision.
- Rules:
- All nonzero digits are significant.
- Zeros between significant digits are significant.
- Zeros to the right of the decimal and to the right of nonzero digits are significant.
- Examples:
- 143 has 3 sf; 0.0032 has 2 sf; 12300 has 3 sf (trailing zeros in whole numbers are not always significant); 3.007 has 4 sf.
- Rounding to a specified number of sf follows: move to the first significant figure, then round the next digit; placeholders fill with zeros as needed.
Estimation
- Estimation is used when exact values are unnecessary; often done by rounding to 1 sf (or 2 sf in some contexts) to check reasonableness.
Indices and logarithmic laws (11.1.1.5)
- Indices (exponents) basics: a^m denotes a multiplied by itself m times; the exponent is the index.
- Index laws (multiplication and powers):
- Law 1: $b^{m} \cdot b^{n} = b^{m+n}$
- Law 2: $(b^{m})^{n} = b^{mn}$
- Law 3: $(ab)^{m} = a^{m} b^{m}$
- Law 4: $\frac{b^{m}}{b^{n}} = b^{m-n}$ (for m ≥ n; division of powers with same base)
- Law 5: $(\frac{a}{b})^{m} = \frac{a^{m}}{b^{m}}$
- Law 6: $a^{0}=1$ for a ≠ 0
- Law 7: $a^{-m}=\frac{1}{a^{m}}$ (negative powers give reciprocals)
- Law 8: Using the quotient rule for exponents: to raise a quotient to a power: $(a/b)^{m} = a^{m}/b^{m}$
Logarithmic laws
- If $x=a^{n}$ , then $\log{a} x = n$ ; equivalently, $a^{\log{a} x}=x$ ; laws include:
- $\log_{a} a = 1$
- $\log{a} (xy) = \log{a} x + \log_{a} y$
- $\log{a} (x^{m}) = m \log{a} x$
- $\log{a} \left(\frac{x}{y}\right) = \log{a} x - \log_{a} y$
- $\log_{a} 1 = 0$
Standard Index Form (SIF) / Scientific notation (11.1.1.6)
- SIF is written as $x = a \times 10^{n}$ with 1 ≤ a < 10; the exponent n indicates the move of the decimal point when converting between standard notation and SIF.
- Examples:
- Large number: $18{,}230{,}000{,}000{,}000 = 1.823 \times 10^{13}.$
- Small number: $0.0027 = 2.7 \times 10^{-3}.$
- Operations in SIF follow index laws for the coefficients and the powers separately.
The Scientific Calculator (11.1.1.7)
- Functions: trigonometric, logarithmic, roots, powers, permutations, combinations, statistics, etc.
- Key features: large display; ability to show input and output; programmable models allow storing formulas and data; memory functions (M+, MR, STO, RCL).
- Common issues: MATH ERROR and SYNTAX ERROR indications; mode settings (DEG vs RAD) affect trigonometric results.
- Practical use: fast computation in higher mathematics; conversion between forms, degree-minute-second conversions, factorials, etc.

11.1.2 UNITS OF MEASUREMENT

11.1.2.1 Metric and Imperial measures (SI units)
- Base SI units (the seven base units):
- Length: meter (m)
- Mass: kilogram (kg)
- Time: second (s)
- Electric current: ampere (A)
- Temperature: kelvin (K) (commonly °C for everyday use, but SI base is K)
- Luminous intensity: candela (cd)
- Amount of substance: mole (mol)
- The metric system uses prefixes to form derived units; the mnemonic given in the text: "Karl Has Developed My Decimal Cravings for Metrics" helps recall prefixes (kilo, hecto, deka, base, deci, centi, milli).
11.1.2.2 Conversion of Metric and Imperial Measures
- Within the Imperial system: common conversions include
- $1\text{ ft} = 12\text{ in}$
- $1\text{ yd} = 3\text{ ft}$
- $1\text{ mi} = 1760\text{ yd}$
- $1\text{ gal} = 4\text{ qt}$
- $1\text{ qt} = 2\text{ pt}$
- Within the Metric system: units scale by powers of 10; moving the decimal point by 1, 2, or 3 places, depending on prefix.
- Between systems: common cross-system conversions are given, e.g., $1\text{ inch} = 2.54\text{ cm}$ ; $1\text{ foot} = 0.3048\text{ m}$ ; $1\text{ liter} = 1000\text{ mL}$ ; $1\text{ kg} \approx 2.2\text{ lb}$ ; $1\text{ gal} \approx 3.785\text{ L}$ (or 4.56 L in some charts depending on context). Practical approach: multiply by the conversion factor and write the appropriate unit.
11.1.2.3 Measuring Devices and Scales
- Length: rulers, meter sticks, yardsticks; common lengths: 12 in (30 cm) ruler; 18 in rulers; tape measures (up to 20 m).
- Mass: balance scales; spring scales; digital scales.
- Volume: graduated cylinders (10 mL, 50 mL, 100 mL) for liquids; measuring cups for cooking.
- Time: clocks (digital/analog).
- Temperature: thermometers (digital/mercury-based).
- Examples and practical notes: digital scales are common in PNG markets; clock-scale mechanisms; interpretation of graduated cylinders.

11.1.3 RATIO AND PROPORTION

11.1.3.1 Ratio
- Definition: a comparison of two quantities with the same unit; written as a:b and read as “a is to b.”
- Examples and conversions:
- Mikha weight 25 kg; father 75 kg. The ratio father to Mikha is $75:25 = 3:1.$
- In a class with 20 boys and 40 girls: ratio of girls to total students, boys to total; total = 60; girls to total = $40:60 = 2:3$ ; boys to total = $20:60 = 1:3.$
11.1.3.2 Proportion
- Definition: equality of two ratios; expressed as $\frac{a}{b}=\frac{c}{d}$ or $a:b=c:d$ ; means means-extremes product equality: $ad = bc$ .
- Example: If 105 lollies cost K35, how many can be bought for K10? Proportion: $105:35 = x:10\Rightarrow 35x = 1050\Rightarrow x=30.$
11.1.3.3 Types of Proportion
- Direct Proportion: both quantities change in the same direction; $\frac{a}{b}=\frac{c}{d} \text{ with } \frac{a}{b} = \frac{c}{d}$ ; equivalently, $ad=bc.$
- Inverse Proportion: one quantity increases, the other decreases; $a:b=c:d \Rightarrow ab = cd$ ; examples include travel time vs speed.
- Partitive (or part-to-whole) Proportion: dividing a whole into parts according to a given ratio; example with a 270 m wire cut in ratio 2:3:4; parts are 60 m, 90 m, 120 m respectively (since total parts = 9; specific parts are proportional to 2,3,4).
11.1.3.4 Scales
- Scale is the ratio between a model’s length and the real length; e.g., 1:100 means 1 cm on the plan equals 1 m on the ground.
- Bar scales and reading maps: identify scale from the bar; example 1:1000; on a plan, 1 cm on the map equals 10 m on the ground if the scale is 1:1000.
- Applications: determining actual dimensions from a drawing, and vice versa; calculating area/length on ground given plan measurements.
11.1.3.5 Practical Applications
- Uses: currency exchange, travel distances, scaling up/down building models, enlargement/reduction, and general conversions.
- Example: currency exchange rate using ratios; distances projected from maps; architectural scaling.

11.1.4 BASIC ALGEBRA

11.1.4.1 Factoring
- Factoring (factorising) expresses a polynomial as a product of factors; prime factorization for numbers, or Highest Common Monomial Factor (HCMF) for polynomials.
- Examples:
- 12 + 4y = 4(3+y) → HCMF is 4.
- x^3 + x^4 + x^2 = x^2(x+1)(1+x) but common factor extraction yields x^2(1+x)(1+x) and then simplify.
- 2x^3 – 6x + 10x^2 = 2x(x^2 – 3 + 5x) (HCMF = 2x).
- 15x^2y^3 + 5x^3y = 5x^2y(xy^2 + x) = 5x^2y(xy^2 + x).
- Factorising simple quadratic trinomials x^2 + bx + c via method of two numbers: find d,e such that d+e=b and de=c; then x^2+bx+c=(x+d)(x+e).
11.1.4.2 Simplifying Algebraic Fractions
- Algebraic fractions are rational expressions; steps to simplify:
- Factor numerator and denominator; cancel common factors; reduce to lowest terms.
- Examples illustrate factoring numerator/denominator and canceling common factors; more complex cases involve factoring polynomials and using cross-cancellation.
11.1.4.3 Solving and Sketching Quadratic Equations
- Quadratic equations: $ax^2+bx+c=0,\; a\neq 0$ .
- Methods:
- Factoring (when possible): e.g., $x^2-5x+6=0\Rightarrow (x-2)(x-3)=0$ , roots ${2,3}$.
- Completing the square; Quadratic Formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$
- Special cases: x^2=81 → x=\pm 9; (x+3)^2=49 → x=4 or x=-10; x^2+8x+16=36 → (x+4)^2=36 → x=2 or x=-10.
- Graphing quadratics: y= ax^2+bx+c yields a parabola; vertex form and intercepts can be found; axis of symmetry; sign of a determines whether it opens up or down.
11.1.4.4 Inequalities
- Inequalities use symbols >, <, \ge, \le; solutions are plotted as regions on a number line; boundary line is solid for \ge, \le and dashed for >, <.
- Examples: solving linear inequalities such as 6x-7 ≤ 5 → x ≤ 2; 4(x-2) > 20 → x > 7; 5-6x ≥ -19 → x ≤ 4.
- When there are two variables, solutions form a region on a graph; shading indicates the feasible set.
Summary of major points (selected highlights from the module)
- A real number includes all numbers on the line; zeros and negatives are valid; distance from zero is absolute value.
- Axioms underpin arithmetic on real numbers; these allow consistent algebraic manipulation.
- Surds, rational/irrational numbers, and logarithms form essential tools for advanced maths; relationships among them are governed by standard laws.
- Indices and logarithms provide compact ways to express and manipulate large and small numbers; rules enable simplification and solving exponents/logarithms problems.
- Units, measurement systems, and conversions are foundational for real-world problems; accurate conversion requires attention to prefixes and conversion factors.
- Ratio, proportion, and scales are essential for comparing quantities, modeling real-world problems (e.g., maps, currency, construction).
- Factoring, algebraic fractions, quadratics, and inequalities form the core tools in solving algebraic problems; multiple techniques are used depending on the structure of the equation.

Connections to context, ethics, and practical relevance

The Secretary’s Message (Module Introduction) emphasizes that Flexible, Open and Distance Education (FODE) aims to broaden access, align with PNG national education goals, and provide alternative pathways for learners; the math content supports practical numeracy and problem-solving relevant to daily life, work, and civic participation.
Real-world relevance: measurement units, conversions, and estimation underpin everyday tasks (cooking, building, travel, finance); ratio and proportion underpin budgeting, currency exchange, and resource allocation; algebra underpins modeling and decision-making.
Practical implications: mastery of sf and estimation aids in error budgeting and safety in engineering and science contexts; understanding SIF and calculators enables efficient handling of large and small numbers in science and economics.
Ethical/philosophical note: the curriculum emphasizes access, equity, and the use of affordable technology to bridge educational gaps; mathematical literacy is linked to informed citizenship and opportunities for social and economic advancement.

Key formulas and representative examples (LaTeX)

Real numbers and sets
- Rational numbers: Q=\left{\frac{a}{b}: a,b\in\mathbb{Z},\; b\neq 0\right}
- Irrational numbers: e.g., $\sqrt{2}, \pi$
Absolute value and opposites
- Absolute value: $|n|$ is the distance of $n$ from 0.
Axiom examples (selected)
- Closure of addition: $a+b\in\mathbb{R}$ for all $a,b\in\mathbb{R}$
- Commutativity of addition: $a+b=b+a$
- Distributivity: $a(b+c)=ab+ac$
Surd relations (conceptual forms)
- $\sqrt{ab}=\sqrt{a}\sqrt{b}\quad (\text{when appropriate})$
- Conjugates: $(a+b)(a-b) = a^2 - b^2$
- Rationalizing denominators (example): $\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$
Index laws
- $b^{m}b^{n}=b^{m+n}$
- $(b^{m})^{n}=b^{mn}$
- $(ab)^{m}=a^{m}b^{m}$
- $\frac{b^{m}}{b^{n}}=b^{m-n}$
- $a^{0}=1$ and $a^{-m}=1/a^{m}$
Logarithms (laws)
- $\log_{a} a = 1$
- $\log{a}(xy) = \log{a} x + \log_{a} y$
- $\log{a}(x^{m}) = m\log{a} x$
- $\log{a}\left(\frac{x}{y}\right) = \log{a} x - \log_{a} y$
- $\log_{a} 1 = 0$
Standard index form (SIF)
- x = a\times 10^{n}, \quad 1\le a<10
Quadratic formula (basic form)
- For $ax^2+bx+c=0, bsp a\neq 0$ , roots are
- $x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}$

Note

This notes packet consolidates essential concepts and procedures from 11.1 Numbers and Applications (Grade 11 Advanced Mathematics, Module 1: Numbers and Applications).
The content above preserves the structure and core ideas of the transcript while presenting them as study notes with explicit formulas in LaTeX.
For your exam, be sure you can reproduce the main laws, apply factoring, solve quadratic equations, work with surds and logarithms, and perform unit conversions and ratio-proportion problems using the methods outlined above.