Standard Form: Writing 166 and Other Numbers in Scientific Notation | 标准形式：将166及其他数字写成科学记数法

📚 Standard Form: Writing 166 and Other Numbers in Scientific Notation | 标准形式：将166及其他数字写成科学记数法

In Edexcel IGCSE Mathematics, standard form (often called scientific notation) is a powerful tool for expressing very large or very small numbers concisely. Take the number 166, for example. Instead of writing it as 166, we can represent it as 1.66 × 10². This simple idea unlocks the ability to work with numbers ranging from the diameter of a galaxy to the size of a virus, all on the same page. Understanding standard form helps you manage calculations in physics, chemistry, and real‑world data without losing track of zeros.

在爱德思IGCSE数学中，标准形式（常称为科学记数法）是简洁表达非常大或非常小数字的强大工具。以数字166为例，我们可以把它写成1.66 × 10²，而不必写成166。这个简单的概念让我们能够在一页纸上处理从星系直径到病毒尺寸的各种数值。掌握标准形式有助于你管理物理、化学和现实世界数据中的计算，不必再为丢失零的个数而烦恼。

1. Introduction to Standard Form | 标准形式简介

Standard form is a way of writing numbers as a product of a number between 1 and 10 (including 1, but not 10) and a power of 10. The general pattern is a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. The letter 'a' is often called the mantissa or coefficient, while 10ⁿ gives the order of magnitude. This notation is universal in scientific and engineering fields.

标准形式是一种将数字写成1到10之间（含1，不含10）的数与10的幂的乘积的方式。一般形式为 a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。字母 a 通常称为尾数或系数，10ⁿ 则给出了数量级。这种记法在科学和工程领域中通用。

For IGCSE, you will encounter standard form in the context of writing numbers, converting between ordinary and standard form, and performing calculations such as multiplication, division, addition, and subtraction. The number 166 is a perfect starting point because it is just above 100, making the exponent small and easy to follow.

在IGCSE考试中，你会在书写数字、在普通形式与标准形式之间转换以及进行加、减、乘、除等计算时遇到标准形式。数字166是一个完美的切入点，因为它略大于100，指数很小，易于理解。

---

2. Why Use Standard Form? | 为什么使用标准形式？

Very large numbers, like 93 000 000 miles (the distance from the Earth to the Sun), contain many zeros and are easy to misread. Written in standard form, that distance becomes 9.3 × 10⁷ miles – much clearer. Conversely, the diameter of a human hair is about 0.000 08 metres, which is neatly expressed as 8 × 10⁻⁵ m. Standard form reduces clutter and helps compare magnitudes at a glance.

像93 000 000英里（地球到太阳的距离）这样的大数包含许多零，容易读错。写成标准形式，这个距离变成 9.3 × 10⁷ 英里，清晰得多。相反，一根头发丝的直径约为0.000 08米，用标准形式可干净地表示为 8 × 10⁻⁵ m。标准形式减少了杂乱，让你一眼就能比较数量级。

In the Edexcel IGCSE exam, standard form questions often test your ability to interpret and calculate with numbers given in this format. Mastering it early saves time and prevents careless mistakes with zeros. When you see 166, writing it as 1.66 × 10² reveals that it is in the hundreds – the exponent 2 tells you the decimal point moves two places.

在爱德思IGCSE考试中，标准形式题目常常考查你理解和计算这种格式数字的能力。尽早掌握它可以节省时间并避免因零而出错。当你看到166时，写成 1.66 × 10² 就表明它在百位上——指数2告诉你小数点移动了两位。

---

3. The General Form: a × 10ⁿ | 一般形式：a × 10ⁿ

The core rule is that the coefficient a must be at least 1 and less than 10. This ensures that every number has a unique standard form representation. The exponent n can be positive, negative, or zero. For a number like 166, we place the decimal point after the first non‑zero digit to get 1.66. Then we ask: “How many places must the decimal point move to return to 166?” The answer is 2 places to the right, so n = 2, giving 1.66 × 10².

核心规则是系数 a 必须至少为1且小于10。这确保了每个数字都有唯一的标准形式表示。指数 n 可以是正数、负数或零。对于像166这样的数字，我们将小数点放在第一个非零数字之后得到1.66。然后问：“小数点需要移动多少位才能恢复成166？”答案是向右移动2位，所以 n = 2，得到 1.66 × 10²。

If a number is less than 1, the exponent is negative. For example, 0.000 45 = 4.5 × 10⁻⁴, because the decimal point must move 4 places to the left to become 0.00045. If a number is between 1 and 10, n = 0; e.g., 4.8 = 4.8 × 10⁰. Remember that 10⁰ = 1, so the number remains unchanged.

如果一个数字小于1，指数为负。例如 0.000 45 = 4.5 × 10⁻⁴，因为小数点需要向左移动4位才能变成0.00045。如果数字在1到10之间，n = 0；例如 4.8 = 4.8 × 10⁰。记住 10⁰ = 1，所以数字保持不变。

---

4. Writing Large Numbers (e.g., 166, 5000) | 书写大数（如166, 5000）

To convert a large number like 166 into standard form, identify the first non‑zero digit (1) and place a decimal point after it: 1.66. Then count the number of digits after that first digit up to the original decimal point (which is understood to be at the end of the whole number). For 166, there are two digits after the 1, so the exponent is 2. Hence 166 = 1.66 × 10².

要将像166这样的大数转换为标准形式，先找到第一个非零数字（1）并在其后加小数点：1.66。然后数一数从该数字之后到原小数点（整数末尾默认有小数点）之间的数字个数。对于166，1后面有两位数字，因此指数为2。所以 166 = 1.66 × 10²。

Let’s try 5000. The first digit is 5, so a = 5.0 (or simply 5). The digits after 5 are three zeros, so the exponent is 3. Therefore 5000 = 5 × 10³. Notice that trailing zeros after the decimal point are not written if the coefficient is an integer, but for consistency we often retain one decimal place: 5.0 × 10³. Edexcel accepts either as long as 1 ≤ a < 10.

试试5000。第一个数字是5，所以 a = 5.0（或简写为5）。5之后有三位数字（都是零），所以指数为3。因此 5000 = 5 × 10³。注意，如果系数为整数，小数点后的尾随零一般不写，但为了一致性我们常保留一位小数：5.0 × 10³。爱德思考试局接受这两种写法，只要满足 1 ≤ a < 10 即可。

Here is a quick reference for some common large numbers and their standard form around 166:

以下是一些常见大数及其标准形式的快速参考（围绕166）：

Ordinary number	Standard form	普通数字	标准形式
100	1 × 10²	100	1 × 10²
166	1.66 × 10²	166	1.66 × 10²
200	2 × 10²	200	2 × 10²
999	9.99 × 10²	999	9.99 × 10²
1000	1 × 10³	1000	1 × 10³

---

5. Writing Small Numbers (e.g., 0.00045) | 书写小数（如0.00045）

When a number is less than 1, the exponent n will be negative. For instance, 0.00166 can be written by moving the decimal point three places to the right to get 1.66, so 0.00166 = 1.66 × 10⁻³. The negative sign indicates division by that power of 10, i.e., 1.66 / 1000.

当数字小于1时，指数 n 为负数。例如 0.00166 可以通过将小数点向右移动三位得到 1.66，因此 0.00166 = 1.66 × 10⁻³。负号表示除以10的幂，即 1.66 / 1000。

Another example: write 0.000 45 in standard form. Start with the first non‑zero digit, 4, so a = 4.5. To go from 4.5 back to 0.00045, the decimal point must move 4 places left, so the exponent is -4. Hence, 0.00045 = 4.5 × 10⁻⁴. Always count the number of places you move the decimal point to obtain a number between 1 and 10 from the original.

另一个例子：将 0.000 45 写成标准形式。从第一个非零数字4开始，所以 a = 4.5。要从4.5回到0.00045，小数点必须向左移动4位，因此指数为-4。所以 0.00045 = 4.5 × 10⁻⁴。始终数一数你需要将原数的小数点移动几位才能得到一个1到10之间的数。

Small number	Standard form	小数	标准形式
0.166	1.66 × 10⁻¹	0.166	1.66 × 10⁻¹
0.0166	1.66 × 10⁻²	0.0166	1.66 × 10⁻²
0.000 45	4.5 × 10⁻⁴	0.000 45	4.5 × 10⁻⁴
0.000 009	9 × 10⁻⁶	0.000 009	9 × 10⁻⁶

---

6. Converting from Standard Form to Ordinary Numbers | 从标准形式转换回普通数字

To go from standard form back to an ordinary number, you reverse the process. For 1.66 × 10², move the decimal point 2 places to the right, filling with zeros as needed: 1.66 → 16.6 → 166. If the exponent is negative, such as 1.66 × 10⁻³, move the decimal point 3 places to the left: 0.00166. This skill is essential when a problem ends with an answer in standard form but asks for it as an ordinary number.

要从标准形式转换回普通数字，只需逆向操作。对于 1.66 × 10²，将小数点向右移动2位，必要时补零：1.66 → 16.6 → 166。如果指数为负，例如 1.66 × 10⁻³，将小数点向左移动3位：0.00166。当题目最终答案以标准形式给出但要求写成普通数字时，这项技能至关重要。

Be careful with numbers like 8.4 × 10⁵. Moving the decimal point 5 places to the right gives 840 000. Some students mistakenly write 8.40000, forgetting that we add zeros to fill the places. Practice with 2.03 × 10⁴ → 20 300 to build confidence. For a small number like 5.67 × 10⁻², move the point 2 left → 0.0567.

处理 8.4 × 10⁵ 这类数字时要小心。将小数点向右移动5位得到 840 000。有些学生错误地写成8.40000，忘了需要补零来填位。通过练习 2.03 × 10⁴ → 20 300 来增强信心。对于像 5.67 × 10⁻² 这样的小数，向左移动2位 → 0.0567。

---

7. Multiplying Numbers in Standard Form | 标准形式下的乘法

When multiplying two numbers in standard form, multiply the coefficients a₁ and a₂, and add the exponents n₁ and n₂. This relies on the index law xᵐ × xⁿ = xᵐ⁺ⁿ. For example, calculate (3 × 10²) × (2 × 10³). First, 3 × 2 = 6. Then, 10² × 10³ = 10²⁺³ = 10⁵. So the product is 6 × 10⁵. However, you must check that the new coefficient remains between 1 and 10. If it does not, adjust it.

当两个标准形式的数字相乘时，将系数 a₁ 和 a₂ 相乘，并将指数 n₁ 和 n₂ 相加。这依赖于指数定律 xᵐ × xⁿ = xᵐ⁺ⁿ。例如，计算 (3 × 10²) × (2 × 10³)。首先，3 × 2 = 6。然后，10² × 10³ = 10²⁺³ = 10⁵。所以乘积是 6 × 10⁵。然而，你必须检查新的系数是否保持在1到10之间。如果不在，需要进行调整。

Work through an example with 166’s coefficient: (1.66 × 10²) × (4 × 10¹). Multiply coefficients: 1.66 × 4 = 6.64. Add exponents: 2 + 1 = 3. The result is 6.64 × 10³, which is already in correct standard form. But if we had (6 × 10²) × (5 × 10³), 6 × 5 = 30, which is not between 1 and 10. So 30 × 10⁵ must be rewritten as 3.0 × 10¹ × 10⁵ = 3.0 × 10⁶. Always adjust the decimal point and increment the exponent by the number of moves.

用166的系数做一个示例：(1.66 × 10²) × (4 × 10¹)。系数相乘：1.66 × 4 = 6.64。指数相加：2 + 1 = 3。结果是 6.64 × 10³，已经是正确的标准形式。但如果我们有 (6 × 10²) × (5 × 10³)，6 × 5 = 30，不在1到10之间。所以 30 × 10⁵ 需要重写为 3.0 × 10¹ × 10⁵ = 3.0 × 10⁶。始终调整小数点，并将指数增加小数点移动的位数。

---

8. Dividing Numbers in Standard Form | 标准形式下的除法

Division follows a similar pattern: divide the coefficients, and subtract the exponent of the divisor from the exponent of the dividend. For two numbers (a × 10ⁿ) ÷ (b × 10ᵐ), the result is (a ÷ b) × 10ⁿ⁻ᵐ. For instance, (9 × 10⁵) ÷ (3 × 10²) = (9 ÷ 3) × 10⁵⁻² = 3 × 10³. Again, ensure the final coefficient is within the range 1 ≤ a < 10.

除法遵循类似的模式：系数相除，被除数的指数减去除数的指数。对于两个数 (a × 10ⁿ) ÷ (b × 10ᵐ)，结果是 (a ÷ b) × 10ⁿ⁻ᵐ。例如 (9 × 10⁵) ÷ (3 × 10²) = (9 ÷ 3) × 10⁵⁻² = 3 × 10³。同样，要确保最终系数在1到10之间。

Consider (1.66 × 10⁻²) ÷ (2 × 10⁻⁵). Divide coefficients: 1.66 ÷ 2 = 0.83. Subtract exponents: (-2) – (-5) = -2 + 5 = 3. So we get 0.83 × 10³. Now 0.83 is less than 1, so we adjust: 0.83 × 10³ = 8.3 × 10⁻¹ × 10³ = 8.3 × 10². The exponent reduces by 1 when we move the decimal point one place to the right. This two‑step adjustment is a common IGCSE requirement.

考虑 (1.66 × 10⁻²) ÷ (2 × 10⁻⁵)。系数相除：1.66 ÷ 2 = 0.83。指数相减：(-2) – (-5) = -2 + 5 = 3。因此我们得到 0.83 × 10³。现在0.83小于1，所以我们调整：0.83 × 10³ = 8.3 × 10⁻¹ × 10³ = 8.3 × 10²。当我们将小数点向右移动一位时，指数减少1。这种两步调整是IGCSE考试的常见要求。

---

9. Adding and Subtracting in Standard Form | 标准形式下的加法和减法

Addition and subtraction require that the numbers have the same exponent (same power of 10). You cannot simply add coefficients if the powers of ten differ. For example, to add 1.66 × 10² (that is 166) and 2 × 10³ (2000), you must first express both with the same exponent. Convert 1.66 × 10² to 0.166 × 10³. Then add the coefficients: 0.166 + 2 = 2.166, so the sum is 2.166 × 10³. Finally, re‑standardise if needed (here it is already fine).

加法和减法要求这些数字具有相同的指数（相同的10的幂）。如果10的幂不同，不能简单地将系数相加。例如，要加 1.66 × 10²（即166）和 2 × 10³（2000），必须先将两者表示为相同指数。将 1.66 × 10² 转换为 0.166 × 10³。然后将系数相加：0.166 + 2 = 2.166，因此和为 2.166 × 10³。最后，如有需要再进行标准化（这里已经满足）。

Alternatively, you can convert everything to ordinary numbers, perform the addition or subtraction, and then convert back to standard form. This method is often safer under exam pressure. For instance, 1.66 × 10² is 166, and 2 × 10³ is 2000, sum = 2166. In standard form, 2166 = 2.166 × 10³. The choice of method is yours, but showing your working clearly earns marks.

或者，你可以将所有数字转换为普通数字，进行加法或减法，然后再转换回标准形式。在考试压力下，这种方法往往更稳妥。例如， 1.66 × 10² 是166，2 × 10³ 是2000，和为2166。用标准形式表示，2166 = 2.166 × 10³。你可以选择方法，但清晰地展示解题步骤能获得分数。

---

10. Using Standard Form in Real Life | 标准形式在现实生活中的应用

Standard form is everywhere in science and economics. The mass of the Earth is about 5.97 × 10²⁴ kg, while an electron’s mass is about 9.11 × 10⁻³¹ kg. Without standard form, comparing these would be a nightmare. Even numbers like 166 appear in real data: a school hall may hold 166 students, but its floor area in mm² could be around 1.66 × 10⁸ mm². Understanding standard form helps you interpret such vastly different scales.

标准形式在科学和经济学中随处可见。地球的质量约为 5.97 × 10²⁴ kg，而一个电子的质量约为 9.11 × 10⁻³¹ kg。没有标准形式，比较这些将是噩梦。即使像166这样的数字也出现在实际数据中：学校礼堂可能容纳166名学生，但其以平方毫米计的面积约为 1.66 × 10⁸ mm²。理解标准形式有助于你解释如此巨大的尺度差异。

In IGCSE exam questions, you might be asked to calculate the number of viruses that fit across a hair or the time light takes to travel a certain distance, all given in standard form. Often

Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）