Standard Form

Standard form (sometimes called scientific notation or standard index form) gives us a way of writing very big and very small numbers using powers of 10.

Why use standard form?

Some numbers are too big or too small to write easily or for your calculator to display at all
- Imagine the number 50⁵⁰, the answer would take 84 digits to write out
- Try typing 50⁵⁰into your calculator, you will see it displayed in standard form
Writing very big or very small numbers in standard form allows us to:
- Write them more neatly
- Compare them more easily
- Carry out calculations more easily
Exam questions could ask for your answer to be written in standard form

How is standard form written?

In standard form numbers are always written in the form $a \times 10^{k}$ where $a$ and $k$ satisfy the following conditions:
- $1 \leq a < 10$
  - So there is one non – zero digit before the decimal point
- $k \in ℤ$
  - So $k$ must be an integer
- $k > 0$ for large numbers
  - How many times $a$ is multiplied by 10
- $k < 0$ for small numbers
  - How many times $a$ is divided by 10

How are calculations carried out with standard form?

Your GDC will display large and small numbers in standard form when it is in normal mode
- Your GDC may display standard form as aEn
  - For example, $2.1 \times 10^{- 5}$ will be displayed as $2.1 E - 5$
  - If so, be careful to rewrite the answer given in the correct form, you will not get marks for copying directly from your GDC
Your GDC will be able to carry out calculations in standard form
- If you put your GDC into scientific mode it will automatically convert numbers into standard form
  - Beware that your GDC may have more than one mode when in scientific mode
  - This relates to the number of significant figures the answer will be displayed in
  - Your GDC may add extra zeros to fill spaces if working with a high number of significant figures, you do not need to write these in your answer
To add or subtract numbers written in the form $a \times 10^{k}$ without your GDC you will need to write them in full form first
- Alternatively you can use 'matching powers of 10', because if the powers of 10 are the same, then the 'number parts' at the start can just be added or subtracted normally
  - For example $(6.3 \times 10^{14}) + (4.9 \times 10^{13}) = (6.3 \times 10^{14}) + (0.49 \times 10^{14}) = 6.79 \times 10^{14}$
  - Or $(7.93 \times 10^{- 11}) - (5.2 \times 10^{- 12}) = (7.93 \times 10^{- 11}) - (0.52 \times 10^{- 11}) = 7.41 \times 10^{- 11}$
To multiply or divide numbers written in the form $a \times 10^{k}$ without your GDC you can either write them in full form first or use the laws of indices

Exam Tip

Your GDC will give very big or very small answers in standard form and will have a setting which will allow you to carry out calculations in scientific notation
Make sure you are familiar with the form that your GDC gives answers in as it may be different to the form you are required to use in the exam

Worked example

Calculate the following, giving your answer in the form $a \times 10^{k}$ , where $1 \leq a < 10$ and $k \in ℤ$ .

3780 \times 200

ii)

(7 \times 10^{5}) - (5 \times 10^{4})

ai-sl-1-1-1-we-1-standard-form-part-ii

iii)

(3.6 \times 10^{- 3}) (1.1 \times 10^{- 5})

Standard Form (DP IB Maths: AI HL)

Revision Note