# Operations with numbers in scientific notation

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

This sub-topic covers how to perform all core arithmetic operations (multiplication, division, addition, subtraction) on numbers written in scientific notation, the standard form for very large and small values common in scientific and real-world contexts.

**Prerequisites:** Laws of exponents for integer powers; Converting between standard form and scientific notation

## Learning objectives

- Perform multiplication, division, addition and subtraction on numbers in scientific notation
- Adjust results to meet the standard form requirement for scientific notation
- Apply operations to solve real-world problems involving large/small values

## Multiplication and Division

For multiplication and division, we use exponent rules to simplify the calculation: multiply/divide the mantissas (the $a$ terms) separately, then add/subtract the exponents of 10 respectively.

**Worked example:** Calculate $(3.2 \times 10^4) \times (2.5 \times 10^2)$ and $(7.2 \times 10^5) \div (1.2 \times 10^3)$, leaving both answers in scientific notation.

1. For multiplication: Group mantissas and powers of 10 separately:
2. $$(3.2 \times 2.5) \times (10^4 \times 10^2)$$
3. Calculate the product of mantissas and add exponents:
4. $$8 \times 10^{4+2} = 8 \times 10^6$$
5. For division: Group terms and divide mantissas, subtract exponents:
6. $$\frac{7.2}{1.2} \times 10^{5-3} = 6 \times 10^2$$
7. Both results are already in correct scientific notation form.

> **tip**
>
> Separating mantissas and powers of 10 avoids confusing exponent rules with arithmetic of the mantissas.

*Calculator:* allowed

## Addition and Subtraction

**Addition/Subtraction Rule** — To add or subtract values in scientific notation, first convert all terms to have the same power of 10, then add or subtract the mantissas, then adjust the final result to correct form if needed.

*Example:* $3 \times 10^2 + 5 \times 10^1 = 30 \times 10^1 + 5 \times 10^1 = 3.5 \times 10^2$

**Worked example:** Calculate $(6.4 \times 10^3) + (2.1 \times 10^2)$, leaving the answer in scientific notation.

1. Step 1: Rewrite terms to share the same exponent. Rewrite $2.1 \times 10^2$ to match exponent 3:
2. $$2.1 \times 10^2 = 0.21 \times 10^3$$
3. Step 2: Add the mantissas, keep the common exponent:
4. $$(6.4 + 0.21) \times 10^3 = 6.61 \times 10^3$$
5. The result is already in correct form, so this is the final answer.

> **warning**
>
> Never add the exponents when adding scientific notation values: this only applies to multiplication.

*Calculator:* allowed

## Adjusting Results to Correct Form

After performing operations, your result will often have a mantissa that is less than 1 or greater than or equal to 10, which does not meet the standard definition of scientific notation. You must adjust it to get full marks.

**Worked example:** Calculate $(5.0 \times 10^3) \times (4.0 \times 10^2)$, leaving the answer in correct scientific notation.

1. Step 1: Perform the initial multiplication:
2. $$(5.0 \times 4.0) \times 10^{3+2} = 20 \times 10^5$$
3. Step 2: The mantissa 20 is greater than 10, so adjust it to between 1 and 10:
4. $$20 = 2.0 \times 10^1$$
5. Step 3: Combine the exponents to get the final result:
6. $$2.0 \times 10^1 \times 10^5 = 2.0 \times 10^6$$

For a mantissa less than 1, reverse the process: multiply the mantissa by 10 and subtract 1 from the exponent. For example, $0.45 \times 10^{-2} = 4.5 \times 10^{-3}$.

*Calculator:* allowed

## Real-World Applications

Scientific notation is used for all types of real-world measurements of large (populations, astronomical distances) and small (atomic masses, microscopic lengths) values.

**Worked example:** The population of a country is $5.2 \times 10^7$ people, and average annual carbon emission per person is $9.5 \times 10^3$ kg. Calculate total national annual emission in scientific notation.

1. Total emission = population × emission per person, so multiply the two values:
2. $$(5.2 \times 9.5) \times (10^7 \times 10^3) = 49.4 \times 10^{10}$$
3. Adjust to correct scientific notation:
4. $$49.4 \times 10^{10} = 4.94 \times 10^{11}$$

Total annual emission is $4.94 \times 10^{11}$ kg.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Adding exponents when adding numbers in scientific notation without matching exponents first
  - Why it fails: This incorrectly applies multiplication rules to addition, leading to large errors in the result
  - Correct: Always convert all terms to the same power of 10 before adding or subtracting mantissas
- **Wrong:** Forgetting to adjust the mantissa after an operation when it is outside $1 \leq a < 10$
  - Why it fails: Exam markers require answers in standard scientific notation, so you lose marks even if the numerical value is correct
  - Correct: Always check your final result and adjust the exponent and mantissa to meet the requirement
- **Wrong:** Adjusting $35 \times 10^4$ to $3.5 \times 10^3$ instead of $3.5 \times 10^5$
  - Why it fails: Mistake in direction of exponent change: dividing the mantissa by 10 requires increasing the exponent by 1
  - Correct: If you divide the mantissa by $10^k$, add $k$ to the exponent; if you multiply the mantissa by $10^k$, subtract $k$ from the exponent
- **Wrong:** Converting $2.1 \times 10^{-3}$ to $0.021 \times 10^{-2}$ when matching exponents for addition
  - Why it fails: Incorrect scaling of the mantissa when changing the exponent value
  - Correct: To increase the exponent by 1, divide the mantissa by 10, so $2.1 \times 10^{-3} = 0.21 \times 10^{-2}$

## Cheatsheet

| Operation | Rule | Example |
| --- | --- | --- |
| Multiplication | $(a \times 10^m)(b \times 10^n) = ab \times 10^{m+n}$, adjust form | $(2 \times 10^3)(3 \times 10^2) = 6 \times 10^5$ |
| Division | $\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}$, adjust form | $\frac{8 \times 10^5}{2 \times 10^2} = 4 \times 10^3$ |
| Add/Subtract | Convert to same exponent, add/subtract mantissas, adjust form | $3 \times 10^2 + 4 \times 10^1 = 3.4 \times 10^2$ |
| Adjustment | $a \geq 10$: $a = a/10 \times 10^1$, +1 to exponent; $a < 1$: $a = 10a \times 10^{-1}$, -1 to exponent | $25 \times 10^3 = 2.5 \times 10^4$; $0.35 \times 10^2 = 3.5 \times 10^1$ |

## What's next

Mastering operations with scientific notation is a foundational skill for all subsequent topics in IB AI HL, where you will regularly work with large and small real-world values in statistics, calculus, and mathematical modelling. A solid understanding of exponent rules here also supports work with logarithms and exponential growth/decay models, which are heavily weighted in both Paper 1 and Paper 2 assessments. Practising these operations now will build accuracy for more complex problems later in the course.

- [Approximation, error and estimation](https://www.owlsprep.com/study/ib-math-ai-hl-u1-approximation-error-and-estimation/)
- [Arithmetic and geometric sequences and series](https://www.owlsprep.com/study/ib-math-ai-hl-u1-arithmetic-and-geometric-sequences-and/)
- [Financial applications of geometric sequences and series](https://www.owlsprep.com/study/ib-math-ai-hl-u1-financial-applications-of-geometric-sequences/)

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