Scientific & Engineering Notation
9 min read
Quick Answer
Engineering notation writes a value as a number between one and nine hundred and ninety-nine multiplied by a power of ten whose exponent is a multiple of three. Scientific notation instead keeps exactly one digit before the decimal point. Engineering notation is preferred in electronics because every exponent maps directly onto an SI prefix.
Intuition
Writing the exponent instead of the zeros
A number like 2400000000 Hz is hard to read and easy to mistype by a digit. Powers of ten fix that. Ten multiplied by itself three times is a thousand, which we write as 10³; the small raised number counts how many times ten was multiplied. A negative exponent counts divisions instead, so 10⁻³ is one thousandth.
Once you can write powers of ten, any value splits into two short pieces: a small readable number, and an exponent that says how big or small it really is. The frequency above becomes 2.4 × 10⁹ Hz — or, using the prefix that means the same thing, 2.4 GHz.
Which exponent you write depends on the convention you follow, and the difference between the two comes down to a single rule.
- Scientific notation always leaves exactly one non-zero digit in front of the decimal point. It is the convention of physics and of calculators.
- Engineering notation instead insists that the exponent be a multiple of three, and lets the number in front run anywhere from one up to nine hundred and ninety-nine.
The second looks like the fussier rule, and it is the one electronics settled on. SI prefixes come in steps of a thousand, so every allowed engineering exponent has a name: 10⁻⁶ is micro, 10³ is kilo, 10⁶ is mega. Write a value in engineering notation and the prefix can be read straight off it, without a further step of arithmetic in between.
Practitioner
Converting, in both directions
To move from a plain decimal to engineering notation, shift the decimal point three places at a time until the number in front is between one and nine hundred and ninety-nine, and count the shifts. Move the point to the left and the exponent goes up; move it right and the exponent goes down. The exponent you land on is the prefix.
| Exponent | Prefix | Reads as |
|---|---|---|
| 10⁹ | G | giga |
| 10⁶ | M | mega |
| 10³ | k | kilo |
| 10⁰ | — | the bare unit |
| 10⁻³ | m | milli |
| 10⁻⁶ | µ | micro |
| 10⁻⁹ | n | nano |
| 10⁻¹² | p | pico |
Worked example — Two datasheet values, both directions
An older datasheet lists a capacitance as 0.000047 F. Shifting the decimal point six places to the right gives an exponent of minus six, which is micro: the same capacitance is 47 µF.
The same page lists a resistance as 2200000 Ω. Shifting six places the other way gives an exponent of plus six, which is mega: 2.2 MΩ. Neither line does any arithmetic on the value itself. Moving the decimal point and adjusting the exponent to match leaves the capacitance and the resistance exactly as the datasheet gave them.
Most conversion slips show up immediately if you check the size of the answer. The two forms must describe the same quantity, so if one of them looks a thousand times larger than the other, a shift was counted in the wrong direction. Saying the result out loud in words — "forty-seven microfarads" — and comparing it against the decimal you started from takes a second, and it catches the error that a silent decimal-point shift never announces.
Calculators and instruments usually offer an ENG display mode that does exactly this, stepping the exponent in threes as you press it; on a scientific calculator it normally sits next to the SCI and FIX modes.
Plain text has no raised digits, so software, netlists and BOM files almost always use E-notation instead of superscripts: 4.7e3 means 4.7 × 10³, and 1E-9 means 10⁻⁹. SPICE netlists, CSV exports and most component databases all speak this dialect.
Decide how many digits you are entitled to write before you write them. A value copied from a multimeter carries only as many significant figures as the instrument actually resolves; padding it with digits your measurement never justified is a claim you cannot support.
Engineer
Exponent arithmetic, significant figures and rounding
Normalised scientific notation is the form m × 10ⁿ where m is at least one and less than ten, and n is an integer. Engineering notation relaxes the range of m to at least one and less than one thousand, and constrains n to a multiple of three. Every value has exactly one normalised scientific form; it has exactly one engineering form too, which is why the mapping onto SI prefixes is unambiguous.
What makes mental estimation possible is that exponents combine on their own, without any reference to the mantissas:
- Multiplying two values multiplies the mantissas and adds the exponents.
- Dividing subtracts the exponents.
- Raising to a power multiplies the exponent by that power, so a square doubles it and a square root halves it.
Order-of-magnitude estimation is built on those rules, and experienced engineers treat it as routine practice: before trusting a calculated result, work out its exponent alone and check that it is the exponent you expected. A power supply answer that lands in kilowatts when you expected milliwatts is wrong by six decades, and you will find that in seconds without touching the mantissa.
Worked example — When the exponent is not a multiple of three
A sensor runs from 3.3 V and draws 250 µA. Multiplying voltage by current gives 0.000825 W.
The natural scientific form of that result has an exponent of minus four — a decade that has no SI prefix at all, so the answer cannot be read off directly. Engineering notation steps the exponent down to minus six and moves the mantissa up to compensate, giving 825 µW, which reads immediately as microwatts. Instruments and datasheets take the same step for the same reason: a reading is only useful if it lands on a decade a prefix can name.
Significant figures are the digits that carry information. Leading zeros never count — they only locate the decimal point. Digits between non-zero digits always count. Trailing zeros to the right of the decimal point count, and that is the part people get wrong: 4.70 kΩ and 4.7 kΩ are different claims, the first asserting knowledge of a digit the second leaves open. Trailing zeros in a bare whole number are genuinely ambiguous, which is one more reason to write values in engineering notation, where the mantissa's digits state the precision without argument.
When you round, round once, at the end. Carrying a rounded intermediate result into the next step propagates the error and — in a long chain — can move the final digit by more than the rounding you thought you were doing. Keep guard digits internally and present the rounded value only in the final line. The tolerance of the measurement or of the component sets how many digits survive that final rounding, which is the subject of measurement accuracy.
Ratios that span many decades — gain, attenuation, signal-to-noise — are usually written logarithmically rather than with an exponent at all, as decibels. AC quantities need a second number altogether, because they carry a phase as well as a magnitude; that is where complex numbers enter, and engineering notation continues to apply to the magnitude alone.
Anyone comparing values in code has one more thing to watch. Binary floating point cannot represent most decimal fractions exactly, so a value entered as a short decimal and the same value reached by arithmetic may differ in the last bits. Comparisons should allow a tolerance rather than test for exact equality.
Common mistakes
- Leaving the exponent at a value that is not a multiple of three — the result is valid scientific notation but has no prefix, so it cannot be read off a scale or written on a schematic without a second conversion.
- Moving the decimal point and the exponent the same way — shifting the point left must increase the exponent. Sanity-check by asking whether the value got bigger or smaller; it should not have changed at all.
- Rounding intermediate results — round once, in the final step. Rounding at every line quietly accumulates error.
- Padding an answer with digits the measurement never justified — a three-digit meter reading does not become more precise by being written with six digits.
- Reading e as the base of natural logarithms in a netlist — in E-notation, 4.7e3 means 4.7 × 10³. The two meanings of the letter live in different contexts and must not be mixed.
Frequently asked questions
What is the difference between scientific and engineering notation?
Scientific notation keeps exactly one non-zero digit before the decimal point and lets the exponent be any integer. Engineering notation forces the exponent to be a multiple of three and lets the mantissa run from one to just under a thousand, so it lines up with SI prefixes.
Why do engineers prefer exponents in multiples of three?
Because SI prefixes step in thousands. An exponent that is a multiple of three always has a prefix name, so the value can be written, spoken and read on an instrument without any further conversion.
What does 4.7e3 mean in a netlist or spreadsheet?
It is plain-text E-notation for 4.7 × 10³, that is 4700. The letter e here means "times ten to the power of", and has nothing to do with the mathematical constant e.
Do trailing zeros count as significant figures?
After a decimal point, yes — they are a deliberate statement of precision. In a bare whole number they are ambiguous, which is exactly why engineering or scientific notation is preferred when precision matters.
Should I round as I go through a multi-step calculation?
No. Keep extra digits through the working and round only the final answer, otherwise each rounding step adds error to the next.
Knowledge check
Write 0.00000015 F in engineering notation and give it a prefix. (Show answer)
Is 47 × 10⁻⁵ F written in engineering notation? (Show answer)
A calculation gives a result whose exponent is 10⁻⁴. What do you do to put it in engineering notation? (Show answer)
A datasheet gives a resistance as 4.70 kΩ. Why is the final zero there? (Show answer)
References
- Bureau International des Poids et Mesures, The International System of Units (SI), 9th edition — prefix definitions and the rules for expressing quantity values.
- International Organization for Standardization, ISO 80000-1: Quantities and units — General — notation, significant figures and rounding conventions.