📚 T-1-1095 IGCSE Maths: Number, Powers and Standard Form | IGCSE 数学：数、幂与标准形式
In IGCSE Mathematics, number work is the foundation for nearly every other topic. This revision article covers key ideas in the number topic, including types of number, prime factorisation, powers and roots, the laws of indices, standard form, rounding, estimation, and upper and lower bounds. Mastering these ideas helps you answer both simple calculation questions and longer problem-solving questions with confidence.
在 IGCSE 数学中，数运算是几乎所有其他专题的基础。本篇复习文章涵盖数与数系、质因数分解、幂与根、指数定律、标准形式、舍入、估算以及上下界等核心考点。掌握这些内容能帮助你自信地应对简单计算题和较长的解题问题。
1. Number Sets and Place Value | 数集与位值
Numbers in IGCSE can be classified into natural numbers (positive whole numbers), integers (positive and negative whole numbers including zero), rational numbers (fractions and terminating or repeating decimals), irrational numbers (non-repeating, non-terminating decimals such as √2 and π), and real numbers (all numbers on the number line). Place value tells us the value of each digit in a number: in 325.47, the 3 is worth 300, the 2 is worth 20, the 5 is worth 5, the 4 is worth 4/10, and the 7 is worth 7/100.
在 IGCSE 中，数字可以分为自然数（正整数）、整数（正整数、负整数和零）、有理数（分数以及有限小数或循环小数）、无理数（无限不循环小数，如 √2 和 π）以及实数（数轴上的所有数）。位值告诉我们一个数中每个数字的值：在 325.47 中，3 表示 300，2 表示 20，5 表示 5，4 表示 4/10，7 表示 7/100。
2. Prime Factorisation | 质因数分解
A prime number has exactly two distinct factors: 1 and itself. Composite numbers can be written as a product of prime numbers. For example, 72 = 2³ × 3². You can find prime factors by using a factor tree or by repeated division by prime numbers. Writing a number in prime factor index form is the first step in many number problems involving HCF and LCM.
质数是恰好有两个不同因数的数：1 和它本身。合数可以写成质数的乘积。例如，72 = 2³ × 3²。你可以使用因数树或不断除以质数的方法来求质因数分解。把数写成质因数指数形式是许多涉及最大公因数（HCF）和最小公倍数（LCM）的题目的第一步。
3. HCF and LCM | 最大公因数与最小公倍数
The highest common factor (HCF) of two or more numbers is the largest number that divides into all of them without a remainder. The lowest common multiple (LCM) is the smallest number that is a multiple of all of them. Using prime factorisation, write each number in index form. For the HCF, multiply the lowest powers of common prime factors; for the LCM, multiply the highest powers of all prime factors that appear.
两个或多个数的最大公因数（HCF）是能够整除所有这些数且没有余数的最大数。最小公倍数（LCM）是所有这些数的倍数中最小的一个。使用质因数分解时，先把每个数写成指数形式。求 HCF 时，取所有公共质因数的最小幂次相乘；求 LCM 时，取出现过的所有质因数的最高幂次相乘。
| Example | Working |
| Find the HCF and LCM of 72 and 90. | 72 = 2³ × 3² 90 = 2 × 3² × 5 HCF = 2¹ × 3² = 18 LCM = 2³ × 3² × 5 = 360 |
This method is reliable for any pair or group of numbers, as long as the prime factorisation is correct. Always check that your HCF divides both original numbers and that both original numbers divide your LCM.
只要质因数分解正确，这种方法对任意一组数都可靠。一定要检查 HCF 是否能整除原来的两个数，以及原来的两个数是否都能整除 LCM。
4. Fractions, Decimals and Percentages | 分数、小数与百分数
Fractions, decimals and percentages are three ways of representing the same proportion. To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100. To convert a percentage to a fraction, write it over 100 and simplify. When adding or subtracting fractions, find a common denominator first. To multiply fractions, multiply the numerators together and the denominators together. To divide fractions, multiply by the reciprocal of the second fraction.
分数、小数和百分数是表示同一比例的三种形式。将分数转换为小数时，用分子除以分母；将小数转换为百分数时，乘以 100；将百分数转换为分数时，写成 100 分之几并化简。分数加减时要先找公分母；分数相乘时分子乘分子、分母乘分母；分数相除时乘以第二个分数的倒数。
For example, 3/5 = 0.6 = 60%. When solving word problems, choose the form that makes the calculation easiest, and remember to simplify your final answer.
例如，3/5 = 0.6 = 60%。解决文字题时，选择最容易计算的形式，并记得化简最终答案。
5. Powers and Roots | 幂与根
A power such as 5³ means 5 × 5 × 5 = 125. The small raised number is the index or exponent, and the big number is the base. Square roots and cube roots reverse powers: √81 = 9 because 9² = 81; ∛64 = 4 because 4³ = 64. Negative bases need careful use of brackets: (−2)² = (−2) × (−2) = 4, but −2² = −(2 × 2) = −4 because the negative sign is not squared.
幂如 5³ 表示 5 × 5 × 5 = 125。较小的上标数是指数或幂次，较大的数是底数。平方根和立方根是幂的逆运算：√81 = 9，因为 9² = 81；∛64 = 4，因为 4³ = 64。负底数需要小心使用括号：(−2)² = (−2) × (−2) = 4，但 −2² = −(2 × 2) = −4，因为负号没有平方。
6. Laws of Indices | 指数定律
The laws of indices allow you to simplify expressions that contain powers without expanding every term. The most important rules are listed below. These laws only work when the base is the same for multiplication and division.
指数定律用于化简含幂的表达式，而不必展开每一项。最重要的规则如下。这些定律仅在乘法或除法中底数相同时适用。
aᵐ × aⁿ = aᵐ⁺ⁿ | aᵐ ÷ aⁿ = aᵐ⁻ⁿ
(aᵐ)ⁿ = aᵐⁿ | a⁰ = 1 | a⁻ⁿ = 1/aⁿ
For example, 2³ × 2⁴ = 2⁷, and 10⁶ ÷ 10² = 10⁴. A fractional power such as a¹⁄ⁿ means the n-th root of a, so 8¹⁄³ = ∛8 = 2. Always use the laws in the correct order and be especially careful with negative and fractional powers.
例如，2³ × 2⁴ = 2⁷，10⁶ ÷ 10² = 10⁴。分数幂如 a¹⁄ⁿ 表示 a 的 n 次方根，所以 8¹⁄³ = ∛8 = 2。务必按正确顺序使用这些定律，并特别注意负指数和分数指数。
7. Standard Form | 标准形式
Standard form is used to write very large or very small numbers clearly. A number in standard form is written as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer. For example, 45 600 = 4.56 × 10⁴ and 0.00032 = 3.2 × 10⁻⁴. To add or subtract, first convert to ordinary numbers or to the same power of 10. To multiply or divide, calculate the A parts separately and apply the laws of indices to the powers of 10.
标准形式用于清晰表示非常大或非常小的数。标准形式写作 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。例如，45 600 = 4.56 × 10⁴，0.00032 = 3.2 × 10⁻⁴。加减时需要先转换为普通数或相同的 10 的幂。乘除时先分别计算 A 部分，再将指数定律应用于 10 的幂。
For example, (3 × 10⁴) × (2 × 10³) = (3 × 2) × 10⁴⁺³ = 6 × 10⁷. On a calculator, standard form may appear as 4.56E4 or 4.56 × 10⁴.
例如，(3 × 10⁴) × (2 × 10³) = (3 × 2) × 10⁴⁺³ = 6 × 10⁷。在计算器上，标准形式可能显示为 4.56E4 或 4.56 × 10⁴。
8. Rounding and Significant Figures | 舍入与有效数字
Rounding is used to shorten a number while keeping it close to the original value. Decimal places count digits after the decimal point. Significant figures count from the first non-zero digit from the left. For example, 0.003 876 to 2 significant figures is 0.0039. When rounding, look at the next digit: if it is 5 or more, round up; if it is less than 5, leave the previous digit unchanged.
舍入用于缩短数字并保持其接近原值。小数位数指小数点后的位数。有效数字从左边第一个非零数字开始计数。例如，0.003 876 保留 2 位有效数字为 0.0039。舍入时看下一位数字：如果是 5 或更大，则进位；如果小于 5，则前一位保持不变。
Always preserve the number of required decimal places or significant figures in the answer unless the question asks for exact form. In multi-step problems, avoid rounding too early to prevent large errors.
除非题目要求精确值，否则答案要保持所要求的小数位数或有效数字。在多步问题中，避免过早舍入，以防止产生较大误差。
9. Estimation | 估算
Estimation helps you check whether a calculated answer is reasonable. Round each number to 1 significant figure before performing the calculation. For example, to estimate (48.2 × 9.7) / 19.6, use (50 × 10) / 20 = 25. This gives a quick check that an exact answer should be close to 25.
估算有助于检查计算答案是否合理。计算前先把每个数保留 1 位有效数字。例如，估算 (48.2 × 9.7) / 19.6 时，可用 (50 × 10) / 20 = 25。这样可以快速检查精确答案是否接近 25。
Estimation is especially useful in multiple-choice questions and when checking calculator output for mistakes. If your exact answer is very different from the estimate, go back and look for an input or calculation error.
估算在选择题和检查计算器输出错误时尤其有用。如果精确答案与估算值相差很大，就要回头检查输入或计算是否有误。
10. Upper and Lower Bounds | 上界与下界
When a measurement is rounded, the true value could lie within an interval. The upper bound is the largest possible value, and the lower bound is the smallest possible value. If a length is 6.4 cm to 1 decimal place, its lower bound is 6.35 cm and its upper bound is 6.45 cm. For calculations involving addition and multiplication, use lower bounds to find the minimum result and upper bounds for the maximum. For subtraction and division, more care is needed: the minimum of a difference is lower bound minus upper bound, and the minimum of a quotient is lower bound divided by upper bound.
当测量值被舍入后，真实值位于某个区间内。上界是可能的最大值，下界是可能的最小值。如果长度为 6.4 cm，精确到 1 位小数，那么下界为 6.35 cm，上界为 6.45 cm。对于加法和乘法计算，用下界求最小值，用上界求最大值。对于减法和除法，需要更加小心：差的最小值是下界减上界，商的最小值是下界除以上界。
For example, if a = 6.4 cm and b = 3.2 cm, both rounded to 1 decimal place, the minimum value of a − b is 6.35 − 3.25 = 3.10 cm, and the maximum value is 6.45 − 3.15 = 3.30 cm.
例如，如果 a = 6.4 cm，b = 3.2 cm，均精确到 1 位小数，那么 a − b 的最小值为 6.35 − 3.25 = 3.10 cm，最大值为 6.45 − 3.15 = 3.30 cm。
11. Common Mistakes and Exam Tips | 常见错误与考试提示
One common mistake is confusing −a² with (−a)². Another is adding fractions without a common denominator or forgetting to simplify. Students also often write standard form with an A value outside the range 1 ≤ A < 10, or mix up decimal places and significant figures. When using index laws, always check that the base is the same before adding or subtracting powers.
一个常见错误是混淆 −a² 和 (−a)²。另一个错误是分数相加时不先找公分母，或者忘记化简。学生也经常把标准形式中的 A 写成不满足 1 ≤ A < 10 的数，或者混淆小数位数和有效数字。使用指数定律时，一定要先检查底数是否相同，再加减指数。
In the exam, show your method clearly, keep exact values during intermediate steps, and only round the final answer. For bound questions, write down the upper and lower bounds before combining them. This structured approach reduces avoidable errors and earns method marks.
考试时，要清晰展示解题过程，中间步骤保持精确值，只对最终答案舍入。对于上下界问题，先写出上界和下界再进行组合。这种有条理的方法可以减少不必要的错误，并获得方法分。
12. Summary | 总结
This topic brings together number classification, prime factorisation, fractions, powers, standard form, rounding, estimation, and bounds. The key skills are conversion between number forms, using index laws correctly, and understanding how rounding creates a range of possible values. Regular practice with calculator and non-calculator questions will help you develop speed and accuracy.
本专题综合了数的分类、质因数分解、分数、幂、标准形式、舍入、估算和上下界。核心技能包括数字形式之间的转换、正确使用指数定律，以及理解舍入如何产生一个可能
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