# Operations with numbers in scientific notation

> IB Mathematics: Applications and Interpretation SL · Unit 1: Number and Algebra
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u1-operations-with-numbers-in-scientific/

This sub-topic covers how to perform addition, subtraction, multiplication and division with numbers expressed in scientific notation, a standard format for very large and very small values common in real-world science and data problems.

**Prerequisites:** [Index laws for integer exponents](https://www.owlsprep.com/study/ib-math-ai-sl-u1-index-laws/); [Significant figures and rounding](https://www.owlsprep.com/study/ib-math-ai-sl-u1-significant-figures/)

## Learning objectives

- Convert between ordinary and correctly formatted scientific notation
- Perform multiplication, division, addition and subtraction with numbers in scientific notation
- Apply these operations to solve real-world problems involving large/small values
- Recognize common errors and avoid them in exam questions

## Review of scientific notation format

**Correct scientific notation** — A standardized representation of numbers that simplifies calculations with very large or very small values, eliminating long strings of leading or trailing zeros.

*Notation:* a \times 10^k, 1 \leq a < 10, k \in \mathbb{Z}

*Example:* $12000 = 1.2 \times 10^4$, $0.00034 = 3.4 \times 10^{-4}$

Before performing any operations, always confirm your numbers are correctly formatted. If the coefficient is not between 1 (inclusive) and 10 (exclusive), adjust the exponent to correct the format first.

**Worked example:** Convert $0.0612 \times 10^{-3}$ to correct scientific notation

1. Original coefficient is 0.0612, which is less than 1. Adjust to get a coefficient between 1 and 10:
2. $$0.0612 = 6.12 \times 10^{-2}$$
3. Combine with the original exponent, using index law $10^a \times 10^b = 10^{a+b}$:
4. $$6.12 \times 10^{-2} \times 10^{-3} = 6.12 \times 10^{-5}$$
5. Confirm 6.12 is between 1 and 10, so this is the final answer.

## Multiplication and division

Multiplication and division follow directly from index laws. For these operations, you work with coefficients and exponents separately, then adjust the result back to correct scientific notation.

> **info**
>
> Rule: $(a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n}$  
> Rule: $(a \times 10^m) \div (b \times 10^n) = (a \div b) \times 10^{m-n}$

**Worked example:** Calculate $(4.2 \times 10^3) \times (2.5 \times 10^2)$, give your answer in scientific notation

1. Multiply the coefficients first: $4.2 \times 2.5 = 10.5$
2. Add the exponents for the powers of 10: $10^3 \times 10^2 = 10^{3+2} = 10^5$
3. Adjust the result to correct scientific notation, since 10.5 > 10:
4. $$10.5 \times 10^5 = 1.05 \times 10^1 \times 10^5 = 1.05 \times 10^6$$
5. Final answer: $1.05 \times 10^6$

**Worked example:** Calculate $(9.6 \times 10^5) \div (1.2 \times 10^{-2})$, give your answer in scientific notation

1. Divide coefficients: $9.6 \div 1.2 = 8$
2. Subtract exponents: $10^{5 - (-2)} = 10^{5 + 2} = 10^7$
3. 8 is between 1 and 10, so no adjustment needed. Final answer: $8 \times 10^7$

## Addition and subtraction

Addition and subtraction follow different rules to multiplication/division. You cannot just add coefficients and add exponents: you must first match the exponents of both numbers before you can add the coefficients.

> **warning**
>
> Matching exponents is non-negotiable for addition/subtraction. This is the step most students get wrong in exams.

**Worked example:** Calculate $(3.6 \times 10^4) - (8.1 \times 10^3)$, give your answer in scientific notation

1. Identify the highest exponent: 4. Rewrite the second term to have exponent 4:
2. $$8.1 \times 10^3 = 0.81 \times 10^4$$
3. Subtract the coefficients, keeping the common exponent:
4. $$(3.6 - 0.81) \times 10^4 = 2.79 \times 10^4$$
5. 2.79 is between 1 and 10, so final answer is $2.79 \times 10^4$

## Real-world applications

Scientific notation is used across all science and social science contexts to represent very large or very small values, from the size of atoms to the distance between galaxies. You will often need to combine operations to solve problems.

**Worked example:** A single grain of sand has a mass of approximately $2.3 \times 10^{-3}$ g. What is the total mass of $4 \times 10^6$ grains of sand?

1. Total mass = mass per grain × number of grains, so we multiply the two values:
2. $$(2.3 \times 10^{-3}) \times (4 \times 10^6)$$
3. Multiply coefficients: $2.3 × 4 = 9.2$. Add exponents: $10^{-3 + 6} = 10^3$.
4. 9.2 is correctly formatted, so total mass is $9.2 \times 10^3$ g, or 9.2 kg.

## Common pitfalls

- **Wrong:** Adding exponents when adding numbers in scientific notation
  - Why it fails: This confuses multiplication rules with addition rules, leading to a result many orders of magnitude larger than the true value
  - Correct: First rewrite both numbers to have the same exponent of 10, then add the coefficients only
- **Wrong:** Forgetting to adjust the coefficient after multiplication/division to get it between 1 and 10
  - Why it fails: Examiners require answers in correct scientific notation, so this will lose marks even if the calculation is mostly correct
  - Correct: After calculating the product or quotient, always check the coefficient and adjust the exponent to meet the 1 ≤ a < 10 rule
- **Wrong:** Incorrect sign when subtracting a negative exponent for division, e.g. $10^5 \div 10^{-2} = 10^3$
  - Why it fails: Subtracting a negative number is equivalent to adding its positive value, but this is often forgotten under exam pressure
  - Correct: Write out the subtraction explicitly: $m - n = 5 - (-2) = 5 + 2 = 7$, so the exponent is 7
- **Wrong:** Adjusting the higher exponent to match the lower exponent, leading to arithmetic errors
  - Why it fails: Working with negative adjustments increases the chance of sign and decimal point errors
  - Correct: Always adjust the number with the smaller exponent to match the larger exponent, to simplify the calculation

## Cheatsheet

| Operation | Rule | Example |
| --- | --- | --- |
| Multiplication | Multiply coefficients, add exponents, adjust $a$ | $(3 \times 10^2)(2 \times 10^3) = 6 \times 10^5$ |
| Division | Divide coefficients, subtract exponents, adjust $a$ | $(6 \times 10^8) \div (2 \times 10^{-3}) = 3 \times 10^{11}$ |
| Add/Subtract | 1. Match exponents 2. Add/subtract coefficients 3. Adjust $a$ | $5.1 \times 10^4 + 4.2 \times 10^3 = 5.52 \times 10^4$ |

## What's next

Operations with scientific notation are a foundational skill for almost all subsequent topics in IB AI SL, from working with exponential growth and decay in population biology to calculating compound interest for large investments, and analyzing large datasets in statistics. Even when you are allowed to use a calculator, being able to estimate the order of magnitude of your answer helps you catch input errors. Mastery of this topic will make all future calculations involving large or small values much faster and more accurate. Next you will move on to learning about approximation and estimation, which builds on scientific notation to help you check the reasonableness of your results in problem solving.

- [Arithmetic sequences and series](https://www.owlsprep.com/study/ib-math-ai-sl-u1-arithmetic-sequences-and-series/)
- [Geometric sequences and series](https://www.owlsprep.com/study/ib-math-ai-sl-u1-geometric-sequences-and-series/)
- [Financial applications of geometric sequences and series](https://www.owlsprep.com/study/ib-math-ai-sl-u1-financial-applications-of-geometric-sequences/)

---

From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/ib-math-ai-sl-u1-operations-with-numbers-in-scientific/