Number Toolkit

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課程筆記

Types of Numbers

Integers are whole numbers (positive, negative, zero). Natural numbers are positive integers (1,2,3,…).
Multiples: a multiple of an integer is formed by multiplying it by another positive integer (e.g., 12 is a multiple of 3).
Factors: a factor divides a number exactly (e.g., 6 is a factor of 18). Find factors using factor pairs or divisibility tests.
Prime numbers have exactly two distinct factors: 1 and itself. First ten primes: 2,3,5,7,11,13,17,19,23,29. 1 is not prime; 2 is the only even prime.
Square numbers: result of multiplying a number by itself (e.g., 1,4,9,16,25,…). Cube numbers: result of multiplying a number by itself twice (e.g., 1,8,27,64,125).
Square root (√) is the inverse of squaring; cube root (∛) is the inverse of cubing. Square roots of non-square numbers are surds (irrational).
Reciprocal of a number is 1 divided by that number; product of a number and its reciprocal equals 1.

Irrational Numbers

A rational number can be written as a fraction a/b (a,b integers, b≠0). Includes integers, terminating and recurring decimals.
An irrational number cannot be written as a simple fraction; its decimal is non-terminating and non-recurring.
Common irrationals: π, √2, √3, √5, etc. Multiplying an irrational by a non-zero rational gives an irrational (e.g., 2π).
√n is irrational if n is not a perfect square. √4=2 is rational; √2 is irrational.

Negative Numbers

Multiplying/dividing: same signs → positive; different signs → negative. E.g., (-12)÷(-4)=3; (-6)×4=-24.
Adding/subtracting: subtracting a negative = adding the positive (e.g., 5-(-3)=8); adding a negative = subtracting the positive (e.g., 7+(-3)=4).
Real-life contexts: temperature (e.g., 3°C cooling by 5°C gives -2°C) and money/debt (e.g., -200 + (-400) = -600).
Use brackets on calculators for negative numbers: (-3)²=9, but -3²=-9.

Mathematical Symbols

Basic operations: + (addition), − (subtraction), × (multiplication), ÷ (division).
Equals and inequality: = (equal), ≠ (not equal), ≈ (approximately), ≡ (identical), > (greater), < (less), ≥ (greater or equal), ≤ (less or equal).
Other symbols: ( ) brackets, ^ or superscript for powers, √ (square root), ∛ (cube root), ± (plus/minus), π (π).

Order of Operations (BIDMAS/BODMAS)

BIDMAS/BODMAS: Brackets, Indices/Orders (powers, roots), Division/Multiplication (left to right), Addition/Subtraction (left to right).
Fractions have invisible brackets around numerator and denominator (e.g., (2+5)/(7-2)). Roots also have invisible brackets (e.g., √(9+16)).
Worked example: (5-3)+2×7² = 2+2×49 = 2+98 = 100.

Addition & Subtraction

Column method: line up digits by place value, add/subtract right to left, carry/borrow as needed.
For subtraction, if top digit is smaller, borrow 10 from the next column to the left.
Alternative subtraction: counting up (e.g., 673-289: add 1→290, +10→300, +300→600, +73→673; total 384).
Key words: sum/total (addition), difference/take away (subtraction). Estimate to check answers.

Multiplication & Division

Column method: multiply each digit of bottom number by top number, use zeros as placeholders, then add results.
Lattice method: draw grid with diagonals, multiply digit pairs, sum diagonals.
Grid method: split numbers by place value, multiply in grid cells, sum all cells.
Short division (bus stop): for dividing by single digit; work left to right, carry remainders.
Dividing by powers of 10 shifts digits right (e.g., 380÷10=38). Factorising can simplify division (e.g., 1008/28 = 252/7 = 36).

Operations with Decimals

Add/subtract decimals: line up decimal points, use zeros as placeholders, then use column method.
Multiply decimals: convert to integers by multiplying by powers of 10, multiply, then divide by same powers. Or ignore decimal, multiply, then place decimal using estimation.
Divide decimals: write as fraction, multiply numerator and denominator by powers of 10 to make integers, divide, then adjust by dividing/multiplying back.
Always estimate to check decimal placement (e.g., 1.57×0.78 ≈ 2×1=2, so answer ~1.2246).

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練習題

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1.Write down the integer values of x that satisfy the inequality –2 ≤ x < 2.
Easy
- A–2, –1, 0, 1
- B–2, –1, 0, 1, 2
- C–1, 0, 1
- D–2, –1, 0
2.Write two hundred thousand and seventeen in figures.
Easy
- A200 017
- B200 170
- C20 017
- D200 000 017
3.Write 15 060 in words.
Easy
- AFifteen thousand and sixty
- BFifteen thousand sixty
- COne hundred fifty thousand sixty
- DFifteen thousand six hundred
4.From the list 3.56, 5, 196, 8, 7, 12, write down a number that is a multiple of 3.
Easy
- A12
- B5
- C7
- D8
5.From the list 3.56, 5, 196, 8, 7, 12, write down a number that is a cube number.
Easy
- A8
- B5
- C12
- D196
6.From the list 3.56, 5, 196, 8, 7, 12, write down a number that is a prime number.
Easy
- A5
- B8
- C12
- D196
7.Write down a prime number between 50 and 60.
Easy
- A53
- B51
- C55
- D57
8.Write down a prime number between 20 and 30.
Easy
- A23
- B21
- C25
- D27

複習筆記

## Number Toolkit - Night Before Exam Summary ### Types of Numbers - **Integers**: whole numbers (positive, negative, zero) e.g. -3, 0, 2 - **Natural numbers**: positive integers (1,2,3,...) - 0 not included - **Multiples**: result of multiplying an integer by a positive integer. e.g. 12 is a multiple of 3 (3×4) - **Factors**: numbers that divide exactly with no remainder. e.g. factors of 18: 1,2,3,6,9,18. Use divisibility tests (2: last digit even; 3: sum of digits divisible by 3; 5: last digit 0 or 5; etc.) - **Prime numbers**: exactly two distinct factors (1 and itself). First ten: 2,3,5,7,11,13,17,19,23,29. 1 is not prime. 2 is the only even prime. - **Square numbers**: result of number × itself. First 15: 1,4,9,16,25,36,49,64,81,100,121,144,169,196,225 - **Cube numbers**: result of number × itself × itself. First 5: 1,8,27,64,125. Also 10³=1000 - **Square roots**: inverse of squaring. √49 = 7. Both positive and negative roots exist (e.g. ±√25 = ±5). √ of non-square is a surd (irrational). - **Cube roots**: inverse of cubing. ∛64 = 4. - **Reciprocal**: 1 divided by the number. e.g. reciprocal of 3 is 1/3. Any number × its reciprocal = 1. ### Irrational Numbers - Cannot be written as a fraction a/b (a,b integers, b≠0). - Non-terminating, non-recurring decimals. - Examples: π, √2, √3, √5. Also 2π, 3√2 are irrational. - √2 × √2 = 2 (rational) - be careful! ### Negative Numbers - Same signs → positive: (-12)÷(-4)=3, (-6)×(-4)=24 - Different signs → negative: (-12)÷4=-3, 6×(-4)=-24 - Adding a negative = subtracting positive: 7+(-3)=7-3=4 - Subtracting a negative = adding positive: 5-(-3)=5+3=8 - Use brackets on calculator: (-3)² = 9, not -9. ### Mathematical Symbols - = equal, ≠ not equal, ≈ approximately, ≡ identical - > greater than, < less than, ≥ greater than or equal, ≤ less than or equal - Brackets ( ), powers (indices), √ square root, ∛ cube root, ± plus minus, π pi ### Order of Operations (BIDMAS/BODMAS) 1. Brackets 2. Indices (powers, roots) 3. Division and Multiplication (left to right) 4. Addition and Subtraction (left to right) - Fractions have invisible brackets: (2+5)/(7-2) - Roots have invisible brackets: √(9+16) ### Addition & Subtraction - Column method: align digits by place value, carry/borrow as needed. - Words: sum (add), difference (subtract). - Estimate to check: e.g. 3985+1273 ≈ 4000+1000=5000. ### Multiplication & Division - Methods: column, lattice, grid, repeated addition. - Short division (bus stop) for dividing by single digit. - Dividing by powers of 10 shifts digits: 380÷10=38.0, 45÷100=0.45. - Words: product (multiply), quotient (divide). ### Operations with Decimals - Add/subtract: line up decimal points, use zeros as placeholders. - Multiply: ignore decimals, multiply as integers, then place decimal point using estimation (e.g. 1.3×2.3≈3 → answer 2.99). - Divide: write as fraction, multiply numerator and denominator by powers of 10 to make integers, divide, then adjust back. ### Key Tips - Know divisibility tests (2,3,4,5,8,9,10). - Remember first 10 primes, first 15 squares, first 5 cubes. - √n is irrational if n is not a perfect square. - Always estimate to check answers.

Integer: A whole number that can be positive, negative, or zero (e.g. -3, 0, 2).
Natural number: A positive integer (1, 2, 3, ...). 0 is not a natural number.
Multiple: A number that can be formed by multiplying a given integer by another positive integer (e.g. 12 is a multiple of 3).
Factor: A number that divides a given integer exactly with no remainder (e.g. 6 is a factor of 18).
Prime number: A number with exactly two distinct factors: 1 and itself. 1 is not prime. 2 is the only even prime.
Square number: The result of multiplying a number by itself (e.g. 4 = 2×2).
Cube number: The result of multiplying a number by itself twice (e.g. 8 = 2×2×2).
Square root: The value that when multiplied by itself gives the original number (e.g. √49 = 7).
Cube root: The value that when multiplied by itself twice gives the original number (e.g. ∛64 = 4).
Reciprocal: The result of dividing 1 by a number. Any number multiplied by its reciprocal equals 1.
Rational number: A number that can be written as a fraction a/b where a and b are integers and b ≠ 0. Includes all terminating and recurring decimals.
Irrational number: A number that cannot be written as a fraction a/b. A non-terminating, non-recurring decimal. Examples: π, √2.
Surd: A square root that is not an integer (e.g. √3). Surds are irrational numbers.
BIDMAS/BODMAS: Order of operations: Brackets, Indices (Orders), Division and Multiplication (left to right), Addition and Subtraction (left to right).
Integer values: Whole numbers that satisfy a condition (e.g. integers x such that -2 ≤ x < 2 are -2, -1, 0, 1).

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