What scientific notation is
Scientific notation writes any number as a value between 1 and 10 multiplied by a power of ten:
a × 10n where 1 ≤ |a| < 10
So 123,000,000 becomes 1.23 × 10⁸ and 0.000000123 becomes 1.23 × 10⁻⁷.
In code and calculators this is usually written with e: 1.23e8 and 1.23e-7.
The value is not just brevity. It makes the magnitude immediately visible, makes comparison trivial, and makes significant figures explicit in a way that a string of zeros does not.
Significant figures and why zeros are ambiguous
Written as 123,000,000, it is impossible to tell whether the measurement is precise to three figures or nine. Scientific notation removes the ambiguity: 1.23 × 10⁸ states three significant figures, while 1.23000000 × 10⁸ states nine.
The rules for counting significant figures:
- Non-zero digits always count.
- Zeros between non-zero digits count — 1002 has four.
- Leading zeros never count — 0.0012 has two.
- Trailing zeros after a decimal point count — 1.200 has four.
- Trailing zeros in a whole number are ambiguous, which is exactly the problem scientific notation solves.
Engineering notation
A variant that constrains the exponent to multiples of three, so it maps directly onto the SI prefixes.
Where scientific notation gives 1.23 × 10⁵, engineering notation gives 123 × 10³ — which reads immediately as 123 kilo-something. Similarly 45,000,000 becomes 45 × 10⁶, or 45 mega.
This is why engineers and electronics work almost exclusively in this form: 4.7 kΩ, 100 µF, 2.4 GHz. The exponent is doing the same job as the prefix.
The prefixes run: kilo 10³, mega 10⁶, giga 10⁹, tera 10¹² going up; and milli 10⁻³, micro 10⁻⁶, nano 10⁻⁹, pico 10⁻¹² going down.
Floating point and why 0.1 + 0.2 is not 0.3
Computers store real numbers in a binary form of scientific notation, defined by IEEE 754. A double-precision value uses 1 bit for the sign, 11 for the exponent, and 52 for the significand.
The consequence is that decimal fractions which are not sums of powers of two cannot be represented exactly — just as 1/3 cannot be written exactly in decimal. 0.1 in binary is a repeating fraction, so it is stored as the nearest representable value.
This is why 0.1 + 0.2 evaluates to 0.30000000000000004 in most languages. It is not a bug in the language; it is a direct consequence of finite binary representation.
The practical rules: never compare floats with ==, use a small tolerance instead; and never store money as a float — use integer minor units, or a decimal type.
Doing arithmetic in scientific notation
Multiplication: multiply the coefficients and add the exponents. (2 × 10³) × (3 × 10⁴) = 6 × 10⁷.
Division: divide the coefficients and subtract the exponents.
Addition and subtraction require the exponents to match first. (2 × 10³) + (3 × 10⁴) becomes (0.2 × 10⁴) + (3 × 10⁴) = 3.2 × 10⁴.
After any operation, renormalise so the coefficient falls between 1 and 10 — and round to the number of significant figures justified by your least precise input, since arithmetic cannot create precision that was not in the measurements.
Frequently asked questions
What is scientific notation?
Writing a number as a coefficient between 1 and 10 multiplied by a power of ten. 123,000,000 becomes 1.23 times 10 to the 8th, usually written 1.23e8 in code.
What is the difference between scientific and engineering notation?
Engineering notation restricts the exponent to multiples of three so it maps onto SI prefixes. 1.23e5 in scientific notation is 123e3 in engineering notation, which reads as 123 kilo.
Why does 0.1 + 0.2 not equal 0.3?
Because computers store numbers in binary floating point, and 0.1 is a repeating fraction in binary. The nearest representable value is used, so the sum comes to 0.30000000000000004.
How many significant figures does 1200 have?
It is ambiguous, which is exactly what scientific notation fixes. Writing 1.2e3 states two significant figures while 1.200e3 states four.