Mastering Standard Form | 掌握标准形（科学记数法）

📚 Mastering Standard Form | 掌握标准形（科学记数法）

Standard form (also called scientific notation) is a compact way to write very large or very small numbers. In IGCSE Mathematics, you must be able to convert between ordinary numbers and standard form, and perform arithmetic using standard form. This guide covers every skill you need for the exam.

标准形（又称科学记数法）是一种简洁书写极大或极小数字的方式。在 IGCSE 数学中，你必须在普通数字与标准形之间灵活转换，并运用标准形进行计算。本指南涵盖考试所需的全部技能。

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1. What Is Standard Form? | 什么是标准形？

A number is written in standard form when it is expressed as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer. This means the first part must be at least 1 but less than 10, and the power of ten can be positive, negative, or zero.

当一个数字写成 A × 10ⁿ 的形式时，即为标准形，其中 1 ≤ A < 10，且 n 为整数。也就是说，第一部分必须大于或等于 1 且小于 10，而十的幂次可以是正数、负数或零。

For example, 5.2 × 10³ is in standard form because 5.2 is between 1 and 10. However, 52 × 10² is not in standard form because 52 is greater than 10, even though the two expressions have the same value.

例如，5.2 × 10³ 是标准形，因为 5.2 介于 1 和 10 之间。然而，52 × 10² 不是标准形，因为 52 大于 10，尽管这两个表达式数值相等。

A × 10ⁿ, where 1 ≤ A < 10 and n ∈ ℤ

标准形：A × 10ⁿ，其中 1 ≤ A < 10，n 为整数

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2. Writing Large Numbers in Standard Form | 将大数写成标准形

To convert a large number into standard form, place the decimal point after the first non-zero digit. Count how many places the decimal point has moved to the left. This count becomes the positive index n.

要将一个大数转换为标准形，将小数点放在第一个非零数字之后。数一数小数点向左移动了多少位，这个位数就是正的指数 n。

Example: Write 3,700,000 in standard form.

示例：将 3,700,000 写成标准形。

• First non-zero digit is 3, so A = 3.7 | 第一个非零数字是 3，因此 A = 3.7

• Decimal point moves 6 places left | 小数点向左移动 6 位

• Therefore 3,700,000 = 3.7 × 10⁶ | 因此 3,700,000 = 3.7 × 10⁶

Tip: Multiply the decimal part by a power of ten until you reach the original number. This confirms your index is correct.

技巧：将小数部分乘上十的幂次，直到得到原数，即可确认你的指数是否正确。

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3. Writing Small Numbers in Standard Form | 将小数写成标准形

For very small numbers, the decimal point moves to the right, and the index n is negative. Count the number of places the decimal point moves until it sits after the first non-zero digit.

对于非常小的数字，小数点向右移动，指数 n 为负数。数一数小数点移动了多少位，直到它位于第一个非零数字之后。

Example: Write 0.00042 in standard form.

示例：将 0.00042 写成标准形。

• First non-zero digit is 4, so A = 4.2 | 第一个非零数字是 4，因此 A = 4.2

• Decimal point moves 4 places right | 小数点向右移动 4 位

• Therefore 0.00042 = 4.2 × 10⁻⁴ | 因此 0.00042 = 4.2 × 10⁻⁴

0.00042 = 4.2 × 10⁻⁴

Remember: each place you move the decimal point changes the power of ten by one. Moving right gives a negative power; moving left gives a positive power.

请记住：小数点每移动一位，十的幂次就改变一次。向右移动得到负指数，向左移动得到正指数。

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4. Converting Standard Form Back to Ordinary Numbers | 将标准形转换回普通数字

To expand a number from standard form, move the decimal point in A to the right if n is positive, or to the left if n is negative. Add zeros as placeholders where needed.

要将标准形展开为普通数字，当 n 为正数时，将 A 的小数点向右移动；当 n 为负数时，向左移动。需要时用零补足数位。

Standard Form | 标准形	Ordinary Number | 普通数字
2.5 × 10⁴	25,000
8.1 × 10⁻³	0.0081
6 × 10⁶	6,000,000
9.4 × 10⁻²	0.094

With a positive index, write extra zeros after the decimal part until the decimal point reaches the correct position. With a negative index, add leading zeros before the first digit.

当指数为正时，在小数部分之后补写零，直到小数点到达正确位置。当指数为负时，在第一个数字之前添加前导零。

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5. The Laws of Indices in Standard Form | 标准形中的指数法则

Calculations with standard form rely on the laws of indices. The three most important rules are listed below.

标准形计算依赖于指数法则。以下是最重要的三条规则。

• Multiplication: 10ᵃ × 10ᵇ = 10⁽ᵃ⁺ᵇ⁾ | 乘法：10ᵃ × 10ᵇ = 10⁽ᵃ⁺ᵇ⁾

• Division: 10ᵃ ÷ 10ᵇ = 10⁽ᵃ⁻ᵇ⁾ | 除法：10ᵃ ÷ 10ᵇ = 10⁽ᵃ⁻ᵇ⁾

• Power of a power: (10ᵃ)ᵇ = 10ᵃᵇ | 幂的乘方：(10ᵃ)ᵇ = 10ᵃᵇ

When multiplying two numbers in standard form, multiply the A parts together and multiply the powers of ten together separately. Then check that your answer is still in standard form.

当两个标准形数字相乘时，先将 A 部分相乘，再将十的幂次相乘。然后检查结果是否仍然符合标准形。

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6. Multiplying and Dividing in Standard Form | 标准形的乘法与除法

Example: Calculate (3 × 10²) × (4 × 10⁵).

示例：计算 (3 × 10²) × (4 × 10⁵)。

(3 × 4) × 10⁽²⁺⁵⁾ = 12 × 10⁷

The answer 12 × 10⁷ is not in standard form because 12 ≥ 10. Rewrite it as 1.2 × 10⁸.

答案 12 × 10⁷ 不是标准形，因为 12 ≥ 10。应改写为 1.2 × 10⁸。

Example: Calculate (9 × 10⁶) ÷ (3 × 10²).

示例：计算 (9 × 10⁶) ÷ (3 × 10²)。

(9 ÷ 3) × 10⁽⁶⁻²⁾ = 3 × 10⁴

When you get a first part that is 10 or more, or less than 1, adjust it by moving the decimal point one place and changing the index accordingly.

当你得到的第一部分大于等于 10 或小于 1 时，通过移动小数点一位并相应改变指数来调整。

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7. Adding and Subtracting in Standard Form | 标准形的加法与减法

To add or subtract numbers in standard form, first rewrite them so they have the same power of ten. This is similar to adding fractions with the same denominator.

要对标准形数字进行加减，首先将它们改写成相同的十的幂次，这类似于将分数化为相同分母后再加减。

Example: Calculate (4 × 10⁵) + (3 × 10⁴).

示例：计算 (4 × 10⁵) + (3 × 10⁴)。

• Rewrite 3 × 10⁴ as 0.3 × 10⁵ | 将 3 × 10⁴ 改写为 0.3 × 10⁵

• Then (4 + 0.3) × 10⁵ = 4.3 × 10⁵ | 然后 (4 + 0.3) × 10⁵ = 4.3 × 10⁵

Always align the indices before adding or subtracting the A parts. Never add the powers of ten during addition.

加减 A 部分之前，务必先对齐指数。在加法中绝不能把十的幂次相加。

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8. Standard Form on Your Calculator | 计算器上的标准形

Most exam boards allow a scientific calculator. To enter a number like 4.5 × 10⁻⁷, press 4.5 followed by the ×10ˣ key (often labelled “×10ˣ” or “exp”) and then enter -7.

大多数考试局允许使用科学计算器。要输入 4.5 × 10⁻⁷，先按 4.5，再按 ×10ˣ 键（通常标记为 ×10ˣ 或 exp），然后输入 -7。

The display will show something like 4.5E-7, where E stands for “times ten to the power of”. This is standard form using an alternative notation.

屏幕会显示类似 4.5E-7 的形式，其中 E 表示“乘以 10 的幂次”。这是标准形的另一种记法。

When the result of a calculation exceeds the display limit, the calculator automatically switches to standard form. You must be able to interpret this display correctly in your final answer.

当计算结果超出显示位数时，计算器会自动切换为标准形。你必须能够正确理解显示内容，并写出最终答案。

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9. Significant Figures and Standard Form | 有效数字与标准形

Standard form is often used when rounding to a given number of significant figures. The first part A must be written with the correct number of significant figures.

当四舍五入到指定有效数字位数时，常用标准形表示结果。第一部分 A 必须写成正确的有效数字位数。

For example, write 23,846 to 2 significant figures in standard form.

例如，将 23,846 写成保留 2 位有效数字的标准形。

• Round 23,846 to 24,000 | 将 23,846 四舍五入为 24,000

• Write as 2.4 × 10⁴ | 写成 2.4 × 10⁴

For small numbers, the zeros before the first non-zero digit are not significant. They simply position the decimal point and are absorbed into the negative index of ten.

对于小数，第一个非零数字之前的零不是有效数字。它们仅用于定位小数点，并已被吸收到十的负指数中。

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10. Real-World Applications | 实际应用

Scientists use standard form to express quantities such as the mass of the Earth, the speed of light, or the size of a virus. Without standard form, these numbers would be too long to read conveniently.

科学家用标准形表达地球质量、光速或病毒尺寸等量。如果没有标准形，这些数字会过长而难以阅读。

Quantity | 数量	Standard Form | 标准形
Speed of light | 光速	3 × 10⁸ m/s
Mass of Earth | 地球质量	5.97 × 10²⁴ kg
Radius of a hydrogen atom | 氢原子半径	5.3 × 10⁻¹¹ m
Population of the world | 世界人口	8 × 10⁹

Standard form also makes estimation easier. To estimate the value of a product, round each number to one significant figure and then multiply the powers of ten.

标准形也使估算更加容易。要估算乘积的值，将每个数字四舍五入到一位有效数字，然后相乘十的幂次即可。

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11. Common Mistakes and How to Avoid Them | 常见错误及避免方法

Students often lose marks on standard form questions because of small errors. Here are the most frequent pitfalls and the correct approaches.

学生常在标准形题目中因小错误而失分。以下是最常见的陷阱及正确做法。

• Writing A = 0.5 instead of 5. The first part must be at least 1, so 0.5 × 10³ should be written as 5 × 10². | 把 A 写成 0.5 而不是 5。第一部分必须至少为 1，所以 0.5 × 10³ 应写成 5 × 10²。

• Forgetting to adjust the index after moving the decimal point in the answer. | 答案中移动小数点后忘记调整指数。

• Adding powers of ten when adding two standard form numbers. | 两个标准形数字相加时把十的幂次相加。

• Counting zeros incorrectly when converting to standard form. Count carefully in groups of three. | 转换标准形时数错零的个数。建议按三位一组仔细数。

A quick final check: multiply your A value by 10ⁿ and see if you recover the original number. If not, review your decimal movement and index sign.

快速终检：将你的 A 值乘以 10ⁿ，看是否能还原原数。如果不能，请检查你移动小数点的方向和指数符号。

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12. Practice Questions with Solutions | 练习与解答

Work through these five questions on your own, then compare your answers with the solutions below.

请先独立完成以下五道题，再对照下方的参考答案。

Question 1 | 第 1 题: Write 62,000,000 in standard form.

Question 2 | 第 2 题: Write 7.1 × 10⁻⁵ as an ordinary number.

Question 3 | 第 3 题: Calculate (2 × 10³) × (6 × 10⁴), giving your answer in standard form.

Question 4 | 第 4 题: Calculate (8 × 10⁷) ÷ (2 × 10²), giving your answer in standard form.

Question 5 | 第 5 题: Calculate (5 × 10⁴) + (3 × 10⁵), giving your answer in standard form.

Solutions | 参考答案

Q1: 6.2 × 10⁷

Q2: 0.000071

Q3: (2 × 6) × 10⁷ = 12 × 10⁷ = 1.2 × 10⁸

Q4: (8 ÷ 2) × 10⁵ = 4 × 10⁵

Q5: Rewrite 5 × 10⁴ as 0.5 × 10⁵, then 0.5 × 10⁵ + 3 × 10⁵ = 3.5 × 10⁵.

第 1 题：6.2 × 10⁷；第 2 题：0.000071；第 3 题：1.2 × 10⁸；第 4 题：4 × 10⁵；第 5 题：将 5 × 10⁴ 改写为 0.5 × 10⁵，相加得 3.5 × 10⁵。

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By mastering standard form, you can handle huge and tiny numbers confidently across all areas of mathematics and science. Keep practising conversions and mixed calculations until the steps feel automatic. This skill appears frequently in IGCSE non-calculator and calculator papers alike.

掌握标准形后，你就能自信地在数学和科学各领域处理极大极小的数字。请持续练习转换和混合运算，直到步骤变成自然反应。这一技能在 IGCSE 非计算器卷和计算器卷中都频繁出现。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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