Standard Form – Number & Algebra – IB Mathematics AA HL
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 $ 50^{50} $. The answer would take 84 digits to write out.
- Try typing $ 50^{50} $ 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 $ a \times 10^{k} $ where $ a $ and $ k $ satisfy the following conditions:
-
$
1 \le a < 10
$
- So there is one non-zero digit before the decimal point.
-
$
k \in \mathbb{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 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
$
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.
Worked Examples
Calculate each of the following, giving your answer in the form \(a \times 10^k\), where $ 1 \le a < 10 $ and $ k \in \mathbb{Z} $.
Example 1: $3780 \times 200$
Using GDC (scientific mode):
Input directly into GDC as ordinary numbers:
$$
3780 \times 200 = 7.56 \times 10^5.
$$
Your GDC will automatically give the answer in standard form (or something like 7.56E5).
Without GDC:
Calculate the value in ordinary form:
$$
3780 \times 200 = 756000.
$$
Convert this to standard form:
$$
756000 = 7.56 \times 10^5.
$$
$\displaystyle 7.56 \times 10^5$
Example 2: $(7 \times 10^5) - (5 \times 10^4)$
Using GDC (scientific mode):
$$
(7 \times 10^5) - (5 \times 10^4) = 6.5 \times 10^5.
$$
The GDC display might show 6.5E5.
Without GDC:
Convert to ordinary numbers:
$$
7 \times 10^5 = 700000,\quad 5 \times 10^4 = 50000.
$$
Subtract:
$$
700000 - 50000 = 650000.
$$
Convert to standard form:
$$
650000 = 6.5 \times 10^5.
$$
$\displaystyle 6.5 \times 10^5$
Example 3: $(3.6 \times 10^{-3}) \times (1.1 \times 10^{-5})$
Using GDC (scientific mode):
$$
(3.6 \times 10^{-3})\times(1.1 \times 10^{-5}) = 3.96 \times 10^{-8}.
$$
Without GDC:
Multiply the number parts:
$
3.6 \times 1.1 = 3.96.
$
Add the exponents of 10:
$
(-3) + (-5) = -8.
$
So the product is:
$$
3.96 \times 10^{-8}.
$$
$\displaystyle 3.96 \times 10^{-8}$
Example 4: $543000 \times 4500$
Using GDC (scientific mode):
$$
543000 \times 4500 \approx 2.4435 \times 10^9.
$$
Your GDC may display something like 2.4435E9.
Without GDC:
In ordinary form:
$
543000 \times 4500 = 543000 \times (45 \times 100)
= (543000 \times 45) \times 100.
$
Calculate:
$
543000 \times 45 = 24{,}435{,}000,\quad
24{,}435{,}000 \times 100 = 2{,}443{,}500{,}000.
$
Convert to standard form:
$$
2{,}443{,}500{,}000 = 2.4435 \times 10^9.
$$
$\displaystyle 2.4435 \times 10^9$
Example 5: $(1.2 \times 10^6) + (7.8 \times 10^5)$
Using GDC (scientific mode):
$$
(1.2 \times 10^6) + (7.8 \times 10^5) = 1.98 \times 10^6.
$$
The GDC might show 1.98E6.
Without GDC:
Notice
$
(7.8 \times 10^5) = 0.78 \times 10^6.
$
Therefore:
$$
(1.2 \times 10^6) + (0.78 \times 10^6)
= (1.2 + 0.78) \times 10^6
= 1.98 \times 10^6.
$$
$\displaystyle 1.98 \times 10^6$
Example 6: $\frac{4.2 \times 10^7}{2.1 \times 10^3}$
Using GDC (scientific mode):
$$
\frac{4.2 \times 10^7}{2.1 \times 10^3} = 2 \times 10^4.
$$
Without GDC:
Divide the number parts:
$
\frac{4.2}{2.1} = 2.
$
Subtract exponents of 10:
$
10^7 \div 10^3 = 10^{7-3} = 10^4.
$
So the result is:
$$
2 \times 10^4.
$$
$\displaystyle 2 \times 10^4$
Example 7: $(9.6 \times 10^{-4}) - (3.4 \times 10^{-5})$
Using GDC (scientific mode):
$$
(9.6 \times 10^{-4}) - (3.4 \times 10^{-5}) = 9.26 \times 10^{-4}.
$$
Without GDC:
Rewrite with the same power of 10. Notice:
$$
9.6 \times 10^{-4} = 96 \times 10^{-5}.
$$
Then:
$$
(96 \times 10^{-5}) - (3.4 \times 10^{-5})
= (96 - 3.4) \times 10^{-5}
= 92.6 \times 10^{-5}.
$$
Finally,
$
92.6 \times 10^{-5} = 9.26 \times 10^{-4}.
$
$\displaystyle 9.26 \times 10^{-4}$
Example 8: $0.045 \div 500$
Using GDC (scientific mode):
$$
0.045 \div 500 = 9 \times 10^{-5}.
$$
Without GDC:
$
0.045 \div 500 = 0.045 \div (5 \times 10^2).
$
First divide by 5:
$
0.045 \div 5 = 0.009.
$
Then divide by
$
10^2 = 100:
$
$$
0.009 \div 100 = 0.00009 = 9 \times 10^{-5}.
$$
$\displaystyle 9 \times 10^{-5}$
Example 9: $1230000000 \times 0.00049$
Using GDC (scientific mode):
$$
1230000000 \times 0.00049 \approx 6.027 \times 10^5.
$$
Without GDC:
Rewrite in standard form first:
$
1230000000 = 1.23 \times 10^9,\quad
0.00049 = 4.9 \times 10^{-4}.
$
Multiply the number parts:
$
1.23 \times 4.9 = 6.027.
$
Add the powers of 10:
$
10^9 \times 10^{-4} = 10^5.
$
So:
$$
6.027 \times 10^5.
$$
$\displaystyle 6.027 \times 10^5$
Example 10: $\frac{(2.4 \times 10^{-3}) \times (5 \times 10^2)}{4 \times 10^3}$
Using GDC (scientific mode):
$$
\frac{(2.4 \times 10^{-3}) \times (5 \times 10^2)}{4 \times 10^3}
= 3 \times 10^{-4}.
$$
Without GDC:
First multiply
$
(2.4 \times 10^{-3})(5 \times 10^2):
$
$
(2.4 \times 5) \times 10^{(-3 + 2)}
= 12 \times 10^{-1}
= 1.2 \times 10^0
= 1.2.
$
Then divide by
$
4 \times 10^3:
$
$$
\frac{1.2}{4 \times 10^3}
= \left(\frac{1.2}{4}\right) \times 10^{-3}
= 0.3 \times 10^{-3}
= 3 \times 10^{-4}.
$$
$\displaystyle 3 \times 10^{-4}$
Standard Form - Number & Algebra - IB Mathematics AA HL
Standard Form - Number & Algebra - IB Mathematics AA HL
Command of Evidence: Quantitative
