1. Why we need it
Some numbers from the real world:
- Speed of light: $299{,}792{,}458$ m/s.
- Mass of the Sun: $1{,}989{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000$ kg.
- Diameter of a hydrogen atom: $0.000{,}000{,}000{,}053$ m.
- Mass of an electron: $0.000{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}910{,}9$ kg.
Counting the zeros is unbearable, and a single miscount silently destroys the answer. Scientific notation fixes both problems by writing the number as a single small factor times a power of ten — exactly the powers-of-ten structure already at the heart of base-10 place value.
2. The standard form
A number is in scientific notation (or standard form) when it's written as
$$ a \times 10^n $$where $a$ is a real number with $1 \leq |a| < 10$ — exactly one digit (other than zero) to the left of the decimal point — and $n$ is an integer (positive, negative, or zero).
A number written in the form $a \times 10^n$ with $1 \leq |a| < 10$ and integer $n$. The coefficient $a$ holds the digits; the exponent $n$ holds the order of magnitude.
The numbers from the previous section, in standard form:
| Quantity | Decimal | Scientific notation |
|---|---|---|
| Speed of light | $299{,}792{,}458$ | $2.99792458 \times 10^8$ m/s |
| Mass of the Sun | $1.989 \times 10^{30}$ | $1.989 \times 10^{30}$ kg |
| Atom diameter | $0.000{,}000{,}000{,}053$ | $5.3 \times 10^{-11}$ m |
| Electron mass | $\approx 9.109 \times 10^{-31}$ | $9.109 \times 10^{-31}$ kg |
The whole point: the exponent $n$ counts orders of magnitude at a glance. The Sun is $10^{30}$ kg; an electron is $10^{-31}$ kg; the ratio is $\sim 10^{61}$ — a number you can compute mentally in scientific notation but would struggle with in expanded form.
3. Converting to and from standard form
Decimal to scientific notation
Count how many places you'd have to move the decimal point to land between the first digit and the rest. That count is the exponent — positive if you moved left, negative if right.
- $45{,}300{,}000$ — move the point $7$ places left: $4.53 \times 10^7$.
- $0.000{,}042$ — move the point $5$ places right: $4.2 \times 10^{-5}$.
- $8.7$ — already in standard form: $8.7 \times 10^0$.
Scientific notation to decimal
Move the decimal point of $a$ by $|n|$ places: right if $n$ is positive, left if $n$ is negative.
- $3.6 \times 10^4$ — move $4$ places right: $36{,}000$.
- $2.5 \times 10^{-3}$ — move $3$ places left: $0.0025$.
Positive exponent → big number → move the point right. Negative exponent → small number → move the point left. The exponent always tells you the direction and the magnitude of the move.
4. Multiplying and dividing
This is where scientific notation earns its keep. To multiply two numbers in scientific notation, multiply the coefficients and add the exponents:
$$ (a \times 10^m) \cdot (b \times 10^n) = (a \cdot b) \times 10^{m+n}. $$Example:
$$ (3 \times 10^4) \cdot (2 \times 10^5) = 6 \times 10^9. $$Division — divide the coefficients, subtract the exponents:
$$ \frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}. $$ $$ \frac{6 \times 10^8}{2 \times 10^3} = 3 \times 10^5. $$If the multiplication of the coefficients pushes the result outside $[1, 10)$, you have to re-normalize. For example $(5 \times 10^3) \cdot (4 \times 10^2) = 20 \times 10^5 = 2 \times 10^6$ — shift one factor of $10$ from the coefficient into the exponent.
5. Adding and subtracting
Addition is awkward in scientific notation because the exponents don't combine the way they do in multiplication. To add or subtract, both numbers must first be expressed with the same power of ten — then you add or subtract the coefficients.
$$ (3.2 \times 10^4) + (4.1 \times 10^3). $$Convert the second so that it shares the exponent of the first: $4.1 \times 10^3 = 0.41 \times 10^4$. Now the powers match:
$$ 3.2 \times 10^4 + 0.41 \times 10^4 = 3.61 \times 10^4. $$The same trick works for subtraction. The principle is identical to adding fractions: you can't add until the denominators (here, the powers of $10$) match.
Adding a number in $10^4$ to one in $10^8$ leaves the small number essentially invisible — $3 \times 10^4$ added to $5 \times 10^8$ is $5.00003 \times 10^8$. The smaller term is four orders of magnitude beneath the rounding noise of the larger. In physics, this is often used to justify ignoring "small" terms entirely.
6. Significant figures
Scientific notation pairs naturally with the concept of significant figures — the digits in a measurement that actually carry information.
The number $0.00420$ has three significant figures (the leading zeros are placeholders, but the trailing zero after the $42$ is a meaningful digit — it's signalling that the measurement was made to the nearest hundredth of a milliunit, not just the nearest tenth). The number $4200$ is ambiguous — is the $00$ measured precisely, or just trailing? Scientific notation removes the ambiguity:
- $4.2 \times 10^3$ — two significant figures.
- $4.20 \times 10^3$ — three significant figures.
- $4.200 \times 10^3$ — four significant figures.
In scientific notation, every digit in the coefficient is significant by construction. This is one reason scientists use it: it makes precision unambiguous.
The standard convention for results from a calculation: a final answer should have no more significant figures than the least-precise input. Multiplying a $3$-sig-fig number by a $5$-sig-fig number gives a $3$-sig-fig result — the extra precision is illusory.