Scientific notation (standard form)

Scientists and engineers deal with numbers stretching from the size of a galaxy to the size of an atom. To keep notation manageable, they always write a number in scientific notation, also called standard form.

Concept

A number is in scientific notation if it has the form

$k \times 10^{n}, \quad 1 \le k < 10, \quad n \in \mathbb{Z}.$

The $k$ part shows the significant digits, and the $10^n$ part shows the size.

Converting a large number. Move the decimal point left until you get a number between $1$ and $10$. Count the number of places you moved , that is your exponent $n$.

$384{,}000 = 3.84 \times 10^5 \quad (\text{moved } 5 \text{ places left}).$

Converting a small number. Move the decimal point right until you get a number between $1$ and $10$. The exponent is negative, equal to the number of places.

$0.00072 = 7.2 \times 10^{-4} \quad (\text{moved } 4 \text{ places right}).$

Multiplying. Multiply the $k$ parts; add the exponents:

$(2 \times 10^{3}) \times (4 \times 10^{5}) = 8 \times 10^{8}.$

If the resulting $k$ goes above $10$, adjust:

$(5 \times 10^{4}) \times (6 \times 10^{2}) = 30 \times 10^{6} = 3 \times 10^{7}.$

Dividing. Divide the $k$ parts; subtract the exponents:

$\frac{6 \times 10^{8}}{2 \times 10^{3}} = 3 \times 10^{5}.$

Adding/Subtracting. First make the exponents equal. Example:

$3 \times 10^{4} + 4 \times 10^{3} = 30 \times 10^{3} + 4 \times 10^{3} = 34 \times 10^{3} = 3.4 \times 10^{4}.$

Why this matters. With scientific notation:

• Reading huge or tiny quantities is fast , you see the order of magnitude immediately.
• Comparing is easy , compare exponents first; ties broken by the $k$.
• Computing large products by hand becomes feasible.

A few orders of magnitude to know:

Worked examples

Example 1. Express $5{,}600{,}000$ in standard form.

• Move decimal $6$ places left: $5.6$.
• Answer: $5.6 \times 10^{6}$.

Example 2. Express $0.000\,000\,32$ in standard form.

• Move decimal $7$ places right: $3.2$.
• Answer: $3.2 \times 10^{-7}$.

Example 3. Compute $(4 \times 10^{5}) \cdot (3 \times 10^{-2})$.

• Multiply $k$'s: $4 \cdot 3 = 12$.
• Add exponents: $5 + (-2) = 3$.
• Answer: $12 \times 10^{3} = 1.2 \times 10^{4}$.

Example 4. The Sun is $1.5 \times 10^{8}$ km from Earth and light travels $3 \times 10^{5}$ km/s. How long does sunlight take to reach Earth?

• Time $= \dfrac{1.5 \times 10^{8}}{3 \times 10^{5}} = 0.5 \times 10^{3} = 5 \times 10^{2}$ s.
• About $500$ s, or roughly $8.3$ minutes.

Try it yourself

1. Write $63{,}500$ and $0.00089$ in standard form.
2. Write $5.04 \times 10^{4}$ and $2.7 \times 10^{-3}$ in usual form.
3. Compute $(2.5 \times 10^{3}) \cdot (4 \times 10^{2})$.
4. Compute $\dfrac{9 \times 10^{6}}{3 \times 10^{2}}$.
5. Compute $4 \times 10^{5} + 7 \times 10^{4}$.
6. Order from smallest to largest: $5 \times 10^{-3}$, $7 \times 10^{-2}$, $2 \times 10^{-4}$, $9 \times 10^{-3}$.
7. A bacterium weighs about $1 \times 10^{-12}$ g. How much would $1$ million of them weigh?
8. India's population is about $1.4 \times 10^{9}$. If each person uses $10$ litres of water in a day, write the daily consumption in standard form.

Activity

Order-of-magnitude chart. On a long horizontal strip mark powers of $10$ from $10^{-15}$ to $10^{30}$. Place a sticker for each of the quantities in the table above. Add three of your own , your mass in grams, the population of your village or city, the distance from your home to school in metres. Step back and look at how many "decades" separate the smallest from the largest. That distance is what scientific notation tames.