📚 Maths General Knowledge: Core Facts for Classroom PPTs | 数学常识：课堂互动课件核心知识点
This article brings together the core mathematical facts, formulas and examples you need to create an engaging Maths General Knowledge classroom presentation. Each section is written as a slide-ready summary for teachers and students, with clear definitions, key notation and quick examples that can be projected directly onto a classroom screen.
本文汇集了制作数学常识课堂演示所需的核心数学事实、公式和示例。每一节都设计成适合幻灯片使用的摘要，包含清晰定义、关键符号和快速示例，方便教师和学生直接投影到课堂屏幕上使用。
1. Number Systems and Place Value | 数系与位值
Natural numbers (ℕ) start from 1, whole numbers include 0, integers (ℤ) include negative numbers, rational numbers (ℚ) are fractions a/b with b ≠ 0, and irrational numbers cannot be written as exact fractions. Real numbers (ℝ) include all rational and irrational numbers.
自然数（ℕ）从 1 开始，全体数包括 0，整数（ℤ）包括负整数，有理数（ℚ）是满足 b ≠ 0 的分数 a/b，无理数不能写成精确分数。实数（ℝ）包含所有有理数和无理数。
In the number 4 735, the 7 is in the hundreds place, so it represents 700. Place value shows the value of each digit based on its position in the number.
在数字 4 735 中，7 位于百位，所以表示 700。位值显示了每个数字根据其在数字中的位置所代表的值。
Decimal places continue to the right of the decimal point: tenths, hundredths, thousandths. For example, 0.456 has 4 tenths, 5 hundredths and 6 thousandths.
小数点后的数位依次是十分位、百分位、千分位。例如 0.456 中有 4 个十分之一、5 个百分之一和 6 个千分之一。
ℝ = ℚ ∪ ℚ′
实数 = 有理数 ∪ 无理数
2. Real Numbers and the Number Line | 实数与数轴
Every real number can be placed on a number line. Positive numbers lie to the right of 0, negative numbers lie to the left of 0, and the distance from 0 is called the absolute value, written |x|.
每一个实数都可以表示在数轴上。正数位于 0 的右侧，负数位于 0 的左侧，到 0 的距离叫作绝对值，记作 |x|。
Rational numbers can be written as terminating decimals or recurring decimals, such as 0.25 or 0.333... Irrational numbers cannot be written as exact fractions, and common examples include √2, π and e.
有理数可以写成有限小数或循环小数，例如 0.25 或 0.333...。无理数不能写成精确分数，常见例子包括 √2、π 和 e。
ess than b, a > b means a is greater than b, and a ≤ b means a is less than or equal to b.
大小关系在课堂讲解中非常重要：a < b 表示 a 小于 b，a > b 表示 a 大于 b，a ≤ b 表示 a 小于或等于 b。
Between any two different real numbers there is always another real number. This fact helps students understand that the number line is continuous, not made of separated points.
任意两个不同的实数之间总还存在另一个实数。这个事实有助于学生理解数轴是连续的，而不是由彼此分离的点组成的。
3. Arithmetic Operations and Order of Operations | 算术运算与运算顺序
utative, so 3 + 5 = 5 + 3 and 4 × 6 = 6 × 4. Subtraction and division are not commutative.
四种基本运算是加法、减法、乘法和除法。加法和乘法满足交换律，所以 3 + 5 = 5 + 3，4 × 6 = 6 × 4。减法和除法不满足交换律。
The order of operations is often remembered as BIDMAS, BODMAS or PEMDAS: Brackets, Indices, Division and Multiplication, then Addition and Subtraction. Work from left to right within the same priority level.
运算顺序通常用 BIDMAS、BODMAS 或 PEMDAS 记忆：括号、指数、除法与乘法，然后是加法与减法。同一优先级内从左到右计算。
14. This single example is useful on a classroom slide because it shows why order matters.
例如，3 + 4 × 2 = 3 + 8 = 11，而不是 14。如果加上括号，(3 + 4) × 2 = 7 × 2 = 14。这个简单例子很适合放在课堂幻灯片上，因为它说明了运算顺序为何重要。
Inverse operations are used to check answers: addition checks subtraction, multiplication checks division. Estimation before calculating also helps students judge whether an answer is reasonable.
逆运算可以用来检查答案：加法检查减法，乘法检查除法。计算前进行估算也有助于学生判断答案是否合理。
4. Factors, Multiples, Primes and Divisibility | 因数、倍数、质数与整除
A factor divides exactly into a number with no remainder. A multiple is the result of multiplying a number by an integer. For example, factors of 12 are 1, 2, 3, 4, 6 and 12, while multiples of 12 include 12, 24, 36 and 48.
因数能整除一个数且没有余数。倍数是一个数乘以整数得到的结果。例如，12 的因数是 1、2、3、4、6 和 12，而 12 的倍数包括 12、24、36 和 48。
A prime number has exactly two factors: 1 and itself. The number 1 is not prime because it has only one factor. The number 2 is the only even prime number.
质数恰好有两个因数：1 和它本身。数字 1 不是质数，因为它只有一个因数。数字 2 是唯一的偶质数。
is is useful for finding the highest common factor (HCF) and lowest common multiple (LCM).
质因数分解把一个数写成质数的乘积。例如，24 = 2³ × 3。这对于求最大公因数（HCF）和最小公倍数（LCM）非常有用。
For 12 and 18, the HCF is 6 and the LCM is 36. Quick divisibility rules also help: a number is divisible by 2 if it is even, by 3 if its digit sum is divisible by 3, by 5 if it ends in 0 or 5, and by 9 if its digit sum is divisible by 9.
对于 12 和 18，最大公因数是 6，最小公倍数是 36。快速整除规则也很有帮助：一个数如果能被 2 整除，则它是偶数；如果各位数字之和能被 3 整除，则能被 3 整除；如果末位是 0 或 5，则能被 5 整除；如果各位数字之和能被 9 整除，则能被 9 整除。
5. Fractions, Decimals, Percentages and Rounding | 分数、小数、百分数与取整
parts are taken. Equivalent fractions represent the same value, such as 1/2 = 2/4 = 5/10.
分数由分子和分母组成。分母表示整体被分成多少等份，分子表示取了多少份。等值分数表示相同的值，例如 1/2 = 2/4 = 5/10。
To simplify a fraction, divide the numerator and denominator by their highest common factor. To add or subtract fractions, use a common denominator. To multiply fractions, multiply numerators and denominators separately.
化简分数时，用分子和分母的最大公因数同时除以它们。加减分数时，先通分。乘分数时，分子与分子相乘，分母与分母相乘。
Percent means "per 100". So 25% een fractions, decimals and percentages is a core classroom skill and should be shown in a three-column table for clarity.
百分数表示“每 100”。因此 25% = 25/100 = 1/4 = 0.25。在分数、小数和百分数之间转换是课堂核心技能，最好用三列表格清晰展示。
Rounding depends on the required accuracy. To round 3.14159 to 2 decimal places, look at the third decimal digit and round up if it is 5 or more, giving 3.14. Significant figures follow a similar idea but count from the first non-zero digit.
取整取决于所需的精确度。把 3.14159 保留两位小数时，看第三位小数，如果大于或等于 5 就进位，得到 3.14。有效数字遵循类似思路，但要从第一个非零数字开始计数。
6. Powers, Roots and Standard Form | 幂、根号与科学计数法
r exponent. The index laws are essential: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, and (aᵐ)ⁿ = aᵐⁿ.
在幂 aⁿ 中，a 是底数，n 是指数。指数定律非常重要：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ。
reciprocal, so a⁻ⁿ = 1/aⁿ. These rules apply in algebra, science and finance calculations.
任何非零数的 0 次幂等于 1，所以 a⁰ = 1。负指数表示倒数，所以 a⁻ⁿ = 1/aⁿ。这些规则适用于代数、科学和金融计算。
Square roots and cube roots reverse powers. For example, √49 = 7 and ∛27 = 3. The rule √ab = √a × √b is useful when simplifying surds, though it requires a and b to be non-negative for square roots.
平方根和立方根是幂的逆运算。例如，√49 = 7，∛27 = 3。化简根式时，√ab = √a × √b 很有用，但平方根要求 a 和 b 非负。
speed of light is approximately 3.00 × 10⁸ m/s, and 0.00045 can be written as 4.5 × 10⁻⁴.
科学计数法把一个数写成 A × 10ⁿ，其中 1 ≤ A < 10，n 是整数。例如，光速约为 3.00 × 10⁸ m/s，0.00045 可以写成 4.5 × 10⁻⁴。
7. Ratio, Proportion and Rates | 比、比例与速率
A ratio compares two or more quantities of the same kind. The ratio 3 : 2 means that for every 3 parts of the first quantity there are 2 parts of the second. Ratios can be simplified by dividing all parts by their highest common factor.
比用于比较两个或多个同类的量。比 3 : 2 表示第一个量有 3 份时，第二个量有 2 份。比可以通过所有项同时除以最大公因数来化简。
the other decreases, often written as y = k/x. The constant k is called the constant of proportionality.
正比例表示两个量同增同减，通常写成 y = kx。反比例表示一个量增加时另一个量减少，通常写成 y = k/x。常数 k 叫作比例常数。
A rate is a quantity per unit of another quantity. Speed is distance divided by time, density is mass divided by volume, and pressure is force divided by area. These formulas are common in both maths and science lessons.
速率是某个量相对于另一个单位量的量。速度等于距离除以时间，密度等于质量除以体积，压强等于力除以面积。这些公式在数学和科学课上都很常见。
ravels 150 km in 2 hours, its average speed is 150 ÷ 2 = 75 km/h. If 4 identical pens cost £6, then 10 pens cost £6 ÷ 4 × 10 = £15.
例如，一辆汽车 2 小时行驶 150 km，平均速度是 150 ÷ 2 = 75 km/h。如果 4 支相同的笔花费 6 英镑，那么 10 支笔花费 6 ÷ 4 × 10 = 15 英镑。
8. Algebra Basics: Expressions, Equations and Identities | 代数基础：表达式、方程与恒等式
An expression is a mathematical phrase with terms, such as 3x + 5. A term is a part separated by + or - signs, a coefficient is the number multiplying a variable, and a constant is a fixed number.
表达式是由项组成的数学式子，例如 3x + 5。项是由 + 或 - 号分隔的部分，系数是乘以变量的数字，常数是固定不变的数。
such as 2x + 3 = 11. An identity is true for all values of the variable, such as 2(x + 3) ≡ 2x + 6. The symbol ≡ is often used to show an identity.
方程表示两个表达式相等，例如 2x + 3 = 11。恒等式对变量的所有取值都成立，例如 2(x + 3) ≡ 2x + 6。符号 ≡ 常用于表示恒等式。
tive law: a(b + c) = ab + ac. Factorising reverses expansion by taking out common factors.
同类项可以合并：3x + 5x = 8x。括号可以用分配律展开：a(b + c) = ab + ac。因式分解通过提取公因式来逆转向括号展开的过程。
to get 2x = 8, then divide by 2 to get x = 4. Always substitute the answer back to check.
解一元一次方程时，对两边进行相同的运算，直到变量单独留在一边。对于 2x + 3 = 11，先减 3 得 2x = 8，再除以 2 得 x = 4。最后一定要把答案代回原方程检查。
9. Geometry Essentials: Angles, Shapes, Perimeter and Area | 几何基础：角、图形、周长与面积
An acute angle is less than 90°, a right angle is exactly 90°, an obtuse angle is between 90° and 180°, a straight angle is 180°, and a reflex angle is greater than 180° but less than 360°.
锐角小于 90°，直角等于 90°，钝角在 90° 和 180° 之间，平角是 180°，优角大于 180° 且小于 360°。
The angles in a triangle add up to 180°, and the angles in a quadrilateral add up to 360°. Angles on a straight line add up to 180°, and angles around a point add up to 360°. These facts are frequently tested and are ideal for quick classroom recall.
三角形内角和为 180°，四边形内角和为 360°。同一直线上的角相加为 180°，围绕一点的角相加为 360°。这些事实经常被考查，非常适合课堂快速回忆。
th × width, triangle area = ½ × base × height, circle area = πr², and circumference = 2πr.
周长是围绕图形一周的长度。面积是二维图形内部的空间。常见公式包括长方形面积 = 长 × 宽，三角形面积 = ½ × 底 × 高，圆面积 = πr²，圆周长 = 2πr。
Pythagoras' ere c is the hypotenuse. For example, if a = 3 and b = 4, then c² = 9 + 16 = 25, so c = 5.
勾股定理适用于直角三角形：a² + b² = c²，其中 c 是斜边。例如，如果 a = 3，b = 4，那么 c² = 9 + 16 = 25，所以 c = 5。
10. Coordinate Geometry and Graphs | 坐标几何与图像
The Cartesian plane uses two perpendicular axes: the x-axis is horizontal and the y-axis is vertical. A point is written as an ordered pair (x, y), where x is the horizontal coordinate and y is the vertical coordinate.
直角坐标平面使用两条互相垂直的轴：x 轴水平，y 轴竖直。一个点写成有序数对 (x, y)，其中 x 是横坐标，y 是纵坐标。
y₂ - y₁) / (x₂ - x₁). A positive gradient rises from left to right, while a negative gradient falls.
直线的斜率衡量其倾斜程度。对于两点 (x₁, y₁) 和 (x₂, y₂)，斜率 m = (y₂ - y₁) / (x₂ - x₁)。正斜率从左到右上升，负斜率从左到右下降。
s ((x₁ + x₂)/2, (y₁ + y₂)/2), and the distance between them is √((x₂ - x₁)² + (y₂ - y₁)²).
直线可以写成 y = mx + c，其中 m 是斜率，c 是 y 轴截距。两点的中点是 ((x₁ + x₂)/2, (y₁ + y₂)/2)，两点间距离是 √((x₂ - x₁)² + (y₂ - y₁)²)。
have the same gradient. Perpendicular lines have gradients that multiply to -1, so m₁ × m₂
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