Standard Form - Number & Algebra - IB Mathematics AA HL
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
. The answer would take 84 digits to write out. - Try typing
into 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
-
- So there is one non-zero digit before the decimal point.
-
- So
must be an integer.
- So
-
for large numbers- How many times
is multiplied by 10.
- How many times
-
for small numbers- How many times
is divided by 10.
- How many times
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, will be displayed as2.1E - 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 may display standard form as
-
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
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:
- Or:
-
To multiply or divide numbers written in the form
without your GDC, you can either:- Write them in full form first,
- Or use the laws of indices.
Worked Examples
Calculate each of the following, giving your answer in the form
Example 1:
Using GDC (scientific mode):
Input directly into GDC as ordinary numbers:
7.56E5
).
Without GDC:
Calculate the value in ordinary form:
Final Answer:
Example 2:
Using GDC (scientific mode):
6.5E5
.
Without GDC:
Convert to ordinary numbers:
Final Answer:
Example 3:
Using GDC (scientific mode):
Without GDC:
Multiply the number parts:
Add the exponents of 10:
So the product is:
Final Answer:
Example 4:
Using GDC (scientific mode):
2.4435E9
.
Without GDC:
In ordinary form:
Calculate:
Convert to standard form:
Final Answer:
Example 5:
Using GDC (scientific mode):
1.98E6
.
Without GDC:
Notice
Therefore:
Final Answer:
Example 6:
Using GDC (scientific mode):
Without GDC:
Divide the number parts:
Subtract exponents of 10:
So the result is:
Final Answer:
Example 7:
Using GDC (scientific mode):
Without GDC:
Rewrite with the same power of 10. Notice:
Then:
Finally,
Final Answer:
Example 8:
Using GDC (scientific mode):
Without GDC:
First divide by 5:
Then divide by
Final Answer:
Example 9:
Using GDC (scientific mode):
Without GDC:
Rewrite in standard form first:
Multiply the number parts:
Add the powers of 10:
So:
Final Answer:
Example 10:
Using GDC (scientific mode):
Without GDC:
First multiply
Then divide by
Final Answer: