Why Scientific Notation Matters for Extreme Quantities
- Numbers in real life come in wildly different sizes.
- A distance in space might be an enormous number of metres, while the mass of a single atom is a tiny decimal with many zeros.
- Writing these in ordinary decimal form is possible, but it is slow to read, easy to miscount zeros, and awkward to calculate with.
Scientific notation (also called standard form in many countries) is a compact way to write very large and very small numbers so that:
- the size (scale) of the number is immediately visible,
- comparisons are easier,
- multiplication and division become faster (by using the laws of exponents).
Scientific notation
A way to write very large or very small numbers in the form $A\times 10^n$, where $1\le A<10$ and $n$ is an integer.
- Terminology can vary by country.
- In the UK and Australia, standard form usually means scientific notation.
- In some places (for example parts of North America), "standard form" can mean ordinary decimal notation.
The Meaning of the Parts $a$ and $10^n$
- In $a\times 10^{n}$:
- $a$ (the coefficient) is a number between 1 and 10 (including 1 but not 10). It contains the significant digits.
- $10^{n}$ (the power of ten) tells you the scale, and therefore how many places the decimal point has moved.
- A useful way to interpret powers of ten is through place value:
- $10^{3}=1000$ makes numbers 1000 times bigger.
- $10^{-3}=\frac{1}{1000}=0.001$ makes numbers 1000 times smaller.
Exponent
The small raised number in a power, for example the $n$ in $10^{n}$, which tells how many times the base is multiplied by itself. Negative exponents represent reciprocals.
Converting Decimals To Scientific Notation
Step-By-Step Method
- Move the decimal point until the first non-zero digit is just before the decimal point (so the new number is between 1 and 10).
- Count how many places you moved the decimal point.
- Write $\times 10^{n}$, where $n$ is:
- positive if you moved the decimal left (the original number was large),
- negative if you moved the decimal right (the original number was small).
Convert $543700$ to scientific notation.
Solution
Move the decimal 5 places left: $$543700 = 5.437\times 10^{5}$$
Convert $0.000000567$ to scientific notation.
Solution
Move the decimal 7 places right to get a number between 1 and 10, so the exponent is negative: $$0.000000567 = 5.67\times 10^{-7}$$
A common mistake is getting the sign of the exponent wrong.
- Large number (many digits) $\Rightarrow$ decimal moved left $\Rightarrow n>0$.
- Small number (many zeros after the decimal) $\Rightarrow$ decimal moved right $\Rightarrow n<0$.
Deciding If a Number Is Already in Scientific Notation
A number is in scientific notation only if it matches both conditions:
- It has the form $a\times 10^n$.
- The coefficient satisfies $1\le a<10$.
Let's decide whether each of the following is in scientific notation:
- $13.14\times 10^{6}$ is not (because $13.14\ge 10$).
- $9.99\times 10^{-5}$ is.
- $7.24\times 10^{0}$ is (since $10^0=1$).
- $62.05\times 10^{-2}$ is not (coefficient too large).
- $0.37\times 10^{3}$ is not (coefficient too small).
Converting Scientific Notation To Decimal Form
To convert $a\times 10^n$ back to decimal form, shift the decimal point in $a$:
- $n$ places to the right if $n$ is positive,
- $|n|$ places to the left if $n$ is negative.
Write $7.24\times 10^{0}$ in decimal form.
Solution
Since $10^{0}=1$: $$7.24\times 10^{0}=7.24$$
- $10^{0}=1$ is a special case worth remembering.
- It often appears when a quantity is already between 1 and 10.
Exponent Laws You Need for Scientific Notation
Calculations in scientific notation rely on the laws of exponents (for real numbers $a,b$ and integers/exponents $m,n$ where defined):
- Product rule (same base): $a^{m}\times a^{n}=a^{m+n}$
- Quotient rule (same base): $\frac{a^{m}}{a^{n}}=a^{m-n}$
- Power of a power: $(a^{m})^{n}=a^{mn}$
- Power of a product: $(ab)^{n}=a^{n}b^{n}$
- Power of a quotient: $\left(\frac{a}{b}\right)^{n}=\frac{a^{n}}{b^{n}}$
We covered exponent laws in the previous article, so you might revisit it to refresh the memory as well as get some practice.
- Think of $10^{n}$ as a "scale factor".
- Multiplying by $10^{3}$ is like changing units by a factor of 1000.
- Multiplying by $10^{-3}$ is like scaling back down by dividing by 1000.
Doing Calculations in Scientific Notation
Multiplication
- Multiply the coefficients and add the exponents: $$(a\times 10^{m})(b\times 10^{n})=(ab)\times 10^{m+n}$$
- After multiplying, you may need to renormalize (adjust the coefficient back into the interval $1\le a<10$).
Compute $(3.0\times 10^{4})(2.0\times 10^{3})$.
Solution
- Multiply coefficients: $$3.0\times 2.0=6.0$$
- Add exponents: $$10^{4}10^{3}=10^{7}$$
- So $$(3.0\times 10^{4})(2.0\times 10^{3})=6.0\times 10^{7}$$
Compute $(9.0\times 10^{5})(4.0\times 10^{2})$ and write in scientific notation.
Solution
$9.0\times 4.0=36.0$ and $10^{5}10^{2}=10^{7}$, so
$$36.0\times 10^{7}=3.6\times 10^{8}$$
Division
Divide the coefficients and subtract the exponents: $$\frac{a\times 10^{m}}{b\times 10^{n}}=\left(\frac{a}{b}\right)\times 10^{m-n}$$
Compute $\frac{8.0\times 10^{6}}{2.0\times 10^{3}}$.
Solution
$$\frac{8.0}{2.0}=4.0 \text{ and } 10^{6}/10^{3}=10^{3}$$
So the result is $4.0\times 10^{3}$.
- When dividing, students often divide the exponents instead of subtracting them.
- Remember: $$\frac{10^{m}}{10^{n}}=10^{m-n},\text{ not }10^{m/n}.$$
Addition and Subtraction
You can only add or subtract numbers in scientific notation efficiently when they have the same power of ten.
- Rewrite one (or both) numbers so the exponent matches.
- Add/subtract the coefficients.
- Renormalize if needed.
Compute $(3.2\times 10^{5})+(4.5\times 10^{4})$.
Solution
Rewrite $4.5\times 10^{4}$ as $0.45\times 10^{5}$.
Then $$(3.2\times 10^{5})+(0.45\times 10^{5})=(3.65)\times 10^{5}$$
If exponents are very different, addition in decimal form can sometimes be faster, especially when one number becomes a simple decimal (for example, $7.93\times 10^{-1}=0.793$).
Choosing the Best Representation for a Calculation
Both decimal form and scientific notation are useful, the best choice depends on the operation and on the numbers involved.
- Scientific notation is usually better for multiplication/division with very large or very small numbers (use exponent laws).
- Decimal form can be better for addition/subtraction if the numbers have different powers of ten and rewriting them would be messy.
- For mixed calculations, you might switch forms partway through.
- Suppose you want to compute $\frac{a}{f}$ where $a=13.14\times 10^{6}$ and $f=2.01\times 10^{4}$.
- Using scientific notation: $$\frac{13.14\times 10^{6}}{2.01\times 10^{4}}=\left(\frac{13.14}{2.01}\right)\times 10^{2}$$
- This keeps the place-value reasoning simple.
In non-calculator work, show your exponent steps clearly:
- combine the powers of 10 using laws of exponents,
- then deal with the decimal coefficient,
- finish by checking the coefficient is between 1 and 10.
This makes it easier to earn method credit even if you make a small arithmetic slip.
Comparing Size and "Grasping" Extreme Quantities
Scientific notation helps you understand scale.
- The exponent $n$ gives a quick comparison: $10^{8}$ is $10^{3}$ times larger than $10^{5}$.
- If two numbers have the same exponent, compare the coefficients.
Which is larger: $6.2\times 10^{-4}$ or $4.9\times 10^{-3}$?
Solution
Compare exponents: $-3$ is greater than $-4$, so $4.9\times 10^{-3}$ is larger.
- Write $0.00072$ in scientific notation.
- Write $5.03\times 10^{7}$ in decimal form.
- Decide which is in scientific notation: $0.62\times 10^{9}$, $6.2\times 10^{8}$.
- Compute $(2.5\times 10^{3})(4\times 10^{-2})$ and give your answer in scientific notation.
- Explain why $10^{0}=1$ is important when converting.
