Scientific Notation — Writing Very Big and Very Small Numbers

One exponent to rule the zeros

Physics spans absurd scales: the observable universe is ~10^26 meters across; a proton is ~10^-15 meters wide. Writing those as ordinary decimals would drown any equation in zeros. Scientific notation compresses scale into a single exponent:

$$N = a \times 10^n, \qquad 1 \le a < 10$$

The coefficient $a$ carries the significant digits; the exponent $n$ carries the size. Positive $n$ means big ($10^6$ = a million), negative $n$ means small ($10^{-6}$ = a millionth). The sign of $n$ is the fastest scale check in science: it tells you 'how many zeros' at a glance.

Converting back and forth

To convert, slide the decimal point until exactly one non-zero digit sits before it, and count the slides:

• $299{,}792{,}458 \rightarrow 2.99792458 \times 10^8$ (decimal moved 8 left → $n = +8$)
• $0.000000911 \rightarrow 9.11 \times 10^{-7}$ (moved 7 right → $n = -7$)
• $6.02 \times 10^{23}$ already normalized: Avogadro's number, no work needed.

Going back is the reverse: positive exponent, move the decimal right; negative, move it left, padding with zeros. $4.2 \times 10^{-3} = 0.0042$.

Arithmetic with exponents

Powers of ten make arithmetic almost trivial — the exponents do the heavy lifting:

• Multiply: multiply coefficients, add exponents: $(2 \times 10^3)(3 \times 10^4) = 6 \times 10^7$
• Divide: divide coefficients, subtract exponents: $(8 \times 10^9)/(2 \times 10^3) = 4 \times 10^6$
• Powers: raise the coefficient, multiply exponents: $(3 \times 10^2)^3 = 27 \times 10^6 = 2.7 \times 10^7$ (renormalize!)

Mini-example — photons from a bulb: a 60 W bulb emits ~$10^{20}$ visible photons per second. In one hour ($3.6 \times 10^3$ s) that is $10^{20} \times 3.6 \times 10^3 = 3.6 \times 10^{23}$ photons — on the order of a mole of light, from exponents alone.

Order of magnitude: the art of the rough answer

An order of magnitude is a factor of ten — the exponent, roughly. Saying a star is 'five orders of magnitude more massive than a planet' means ~100,000× more massive, no calculator required.

Physicists use this for Fermi estimates: quick, rough calculations that land within a factor of ten. How many piano tuners in a city? How much energy in a hurricane? Break the question into a few estimated powers of ten, multiply, and the exponent gives you the answer's scale. Being wrong by 3× still nails the order of magnitude — and in science, knowing the scale is often knowing the most important thing.

Variables and units

Common mistakes

Real-world applications

• Astronomers write the Sun's mass as 1.99 × 10^30 kg instead of a 31-digit number.

• Particle physicists write the electron mass as 9.11 × 10^-31 kg — impossible to grasp without powers of ten.

• Engineers compare scales instantly: 10^9 Hz vs 10^6 Hz is 'a thousand times faster' at a glance.

• Chemistry and physics share the mole: 6.02 × 10^23 particles, written compactly.