Scientific Notation Explained With Examples

Some numbers are just too unwieldy to write out digit by digit - the mass of a planet, the width of a single atom - so scientists compress them into a small coefficient times a power of 10. Earth's distance from the sun, close to 150,000,000 km, becomes a tidy 1.5 × 10⁸ km once it's written this way, and the same trick works just as well at the other end of the scale for numbers too small to show comfortably in decimal form.

The Format

a × 10ⁿ

Where: 1 ≤ a < 10, and n is any integer

That leading number, a, is the coefficient, and it always has to land between 1 and 10. The exponent n does the real work - it tells you exactly how far, and in which direction, the decimal point has shifted.

Converting Large Numbers to Scientific Notation

Slide the decimal point left until just one non-zero digit remains in front of it. However many places you moved becomes your exponent, and it's positive.

93,000,000
Move decimal 7 places left: 9.3
= 9.3 × 10⁷

450,000
Move decimal 5 places left: 4.5
= 4.5 × 10⁵

Converting Small Numbers to Scientific Notation

For small numbers the process flips: slide the decimal right until one non-zero digit sits in front, and count those moves as a negative exponent.

0.00042
Move decimal 4 places right: 4.2
= 4.2 × 10⁻⁴

0.0000078
Move decimal 6 places right: 7.8
= 7.8 × 10⁻⁶

Converting Back to Standard Form

Reversing the process just means running the decimal point the other way: right for a positive exponent, left for a negative one.

3.6 × 10⁴ → move 4 right → 36,000
2.5 × 10⁻³ → move 3 left → 0.0025

Multiplying in Scientific Notation

Multiplying two numbers in this form takes two easy steps: multiply the coefficients, then add the exponents.

(3 × 10⁴) × (2 × 10³)
= (3 × 2) × 10⁽⁴⁺³⁾
= 6 × 10⁷ = 60,000,000

If that coefficient multiplication pushes the result to 10 or higher, shift it back down a notch and bump the exponent up by one: (12 × 10⁵) becomes (1.2 × 10⁶).

Dividing in Scientific Notation

Division mirrors that process: divide the coefficients, then subtract the exponents.

(8 × 10⁶) ÷ (2 × 10²)
= (8 ÷ 2) × 10⁽⁶⁻²⁾
= 4 × 10⁴

Adding and Subtracting

Addition and subtraction are the odd ones out - both numbers need matching exponents before you can combine them, so you'll usually have to rewrite one first.

(3.5 × 10⁴) + (2.0 × 10³)
= (3.5 × 10⁴) + (0.2 × 10⁴)
= 3.7 × 10⁴ = 37,000

Examples in Science

Quantity	Standard	Scientific Notation
Earth-Sun distance	150,000,000 km	1.5 × 10⁸ km
Mass of electron	0.00000000000000000000000000000091 kg	9.1 × 10⁻³¹ kg
Speed of light	300,000,000 m/s	3 × 10⁸ m/s
Width of human hair	0.00007 m	7 × 10⁻⁵ m

Scientific Notation: Compressing Big and Tiny Numbers