🔬Scientific Notation

Light travels 300,000,000 meters every second. Just counting the zeros is eye-work, and one copy mistake hides forever. Scientists wrestle with such numbers daily, so they agreed on a paper-saving form: scientific notation.

Its engine is exponents: $10^8$ is a 1 followed by 8 zeros.

The standard shape: a × 10ⁿ

Every number can be written

$a \times 10^n$

with $a$ pinned between 1 and 10: $1 \le a < 10$. The speed of light becomes $3 \times 10^8$ m/s; the Earth–Sun distance is about $1.5 \times 10^{11}$ m — slide the decimal point 11 places and you get the original string back.

Why force $1 \le a < 10$? For uniqueness: 5200 is written $5.2 \times 10^3$, the same by everyone, so no offbeat $52 \times 10^2$ can sneak in.

Big numbers: slide the decimal point

Put the point just after the first non-zero digit; the number of places you slid is the exponent.

Small numbers: negative exponents

Numbers below 1 use negative exponents: slide the point right, and the exponent goes negative.

A negative exponent does not make the number negative — it means dividing by a power of ten: $10^{-3} = \frac{1}{1000} = 0.001$, the same story the places in decimals tell.

Multiply and divide the fast way

The real power shows up in arithmetic: multiply the front numbers and add the exponents; divide the front numbers and subtract the exponents.

• $(3 \times 10^8) \times (2 \times 10^5) = 6 \times 10^{13}$: since $3 \times 2 = 6$ and $8 + 5 = 13$;
• $(6 \times 10^7) \div (2 \times 10^4) = 3 \times 10^3$: since $6 \div 2 = 3$ and $7 - 4 = 3$.

No zero-parade required. If the front numbers multiply past 10, normalize once more: $30 \times 10^5 = 3 \times 10^6$.

Check yourself

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Quick quiz

↺ Try again

1. 1. 5200 in scientific notation is…

   52 × 10²5.2 × 10³0.52 × 10⁴

2. 2. 0.008 in scientific notation is…

   8 × 10⁻³8 × 10³0.8 × 10⁻²

3. 3. (2 × 10⁴) × (3 × 10³) = ?

   6 × 10⁷5 × 10⁷6 × 10¹²