Frequency Conversion Explained

Why frequency conversion looks easy and still trips people up

Frequency conversion is one multiplication, but the units run across thirty orders of magnitude — from microhertz (the slow wobble of a planet) up to terahertz (infrared light) — and the prefixes are easy to misread by a factor of a thousand. The frequency converter on Calc Dragon handles every common pair using exact SI prefix factors and exact mechanical relationships, but the answer only means something if the number going in is the number you think it is. This article walks through the maths, the exact constants, the rpm and rad/s edge cases, the link to wavelength, and the reference frequencies that catch errors before they propagate.

The piece covers the hertz-bridge formula every conversion uses, where the SI prefix factors come from (the BIPM SI Brochure, 9th edition), why rpm divides by 60 and why rad/s divides by 2π, the relationship to wavelength via the wave equation c = f × λ, real-world reference frequencies to anchor a figure, and the cases where a unit converter is the wrong tool — modulated signals with a bandwidth as well as a centre frequency, fractional frequencies in music, and angular velocities that are not really frequencies at all.

The math behind every frequency conversion

Every conversion in the frequency converter uses a single intermediate unit: the hertz. Each unit has a "hertz per unit" factor, and the conversion is two multiplications:

result = value × (Hz per source unit) ÷ (Hz per target unit)

So 2.4 GHz expressed in MHz is 2.4 × 10⁹ ÷ 10⁶ = 2 400 MHz. The same hertz bridge handles every pair without needing one factor per source-target combination — only one number per unit is stored, and every other conversion follows from it. This is the standard pattern in scientific software, units libraries, and the SI brochure itself.

The factors used are exact wherever possible. The SI prefix factors are exact decimal powers by definition: 1 kHz = 10³ Hz exactly, 1 MHz = 10⁶ Hz, 1 GHz = 10⁹ Hz, 1 THz = 10¹² Hz, and going the other way 1 mHz = 10⁻³ Hz, 1 μHz = 10⁻⁶ Hz. The mechanical relationships are exact too: 1 rpm = 1/60 Hz exactly (one revolution per 60 seconds), and 1 rad/s = 1/(2π) Hz exactly (one full cycle is 2π radians by definition). The only finite-precision arithmetic is in the floating-point evaluation of 2π, where the relative error is below 10⁻¹⁵. That is far smaller than any frequency measurement you are likely to be checking, and far smaller than the rounding applied to the displayed result.

Worked example: 2.4 GHz Wi-Fi in seven different units

Take the 2.4 GHz Wi-Fi band — channels 1 through 14 of 802.11b/g/n. The frequency converter gives:

• In gigahertz: 2.4 GHz (the headline figure on every router box).
• In megahertz: 2.4 × 10⁹ ÷ 10⁶ = 2 400 MHz. The 2.4 GHz band is officially 2 400–2 483.5 MHz in most jurisdictions.
• In kilohertz: 2.4 × 10⁹ ÷ 10³ = 2 400 000 kHz. Rare in everyday Wi-Fi talk but standard in radio engineering when comparing across very different bands.
• In hertz: 2 400 000 000 Hz, or 2.4 × 10⁹ Hz. The full integer is what physics formulas use.
• In terahertz: 2.4 × 10⁹ ÷ 10¹² = 0.0024 THz. Wi-Fi is far below the THz band, which sits between microwave and infrared light (0.3–3 THz).
• In rpm (if you imagined a spinning disc completing one cycle per Hz): 2.4 × 10⁹ × 60 = 1.44 × 10¹¹ rpm. Physically meaningless for radio, but a good sanity check that the unit conversion is internally consistent.
• In rad/s: 2.4 × 10⁹ × 2π ≈ 1.508 × 10¹⁰ rad/s. This is the angular frequency ω that appears in the time-domain expression of the carrier signal, sin(ωt + φ).

Going the other direction is symmetric: a 433 MHz LoRa signal is 0.433 GHz, 433 000 kHz, or 4.33 × 10⁸ Hz. The converter handles all nine units in a single dropdown, so the source and target can be chosen independently and any pair works without having to know the factor by heart.

Why rpm divides by 60 (and the pole-count gotcha)

Revolutions per minute is the dominant frequency unit for anything that spins — engines, motors, hard drives, turbochargers, centrifuges. The conversion to hertz is one division: rpm ÷ 60 = Hz, because there are 60 seconds in a minute and a hertz is one cycle per second. So a 3 000 rpm engine is at 50 Hz, a 7 200 rpm hard drive is at 120 Hz, and a 100 000 rpm turbocharger is at about 1.67 kHz.

The gotcha is in the other direction, when relating mains frequency to motor speed. A synchronous motor connected to a 50 Hz mains supply does not always spin at 50 × 60 = 3 000 rpm. The synchronous speed depends on the pole count: n = (120 × f) / P, where n is rpm, f is line frequency, and P is the number of magnetic poles in the stator. A 2-pole motor on 50 Hz spins at 3 000 rpm; a 4-pole at 1 500 rpm; a 6-pole at 1 000 rpm; an 8-pole at 750 rpm. Asynchronous (induction) motors run a few percent slower than synchronous speed because of slip, but the same family of round numbers applies. The frequency converter only does the unit conversion; it does not know about pole counts, so always check what spec you are converting.

Why rad/s divides by 2π

Radians per second is the dominant frequency unit in physics and control engineering, where it is called angular frequency and written ω. The conversion is one division: rad/s ÷ 2π = Hz, because one full cycle is 2π radians by definition (it is the angle subtended by a full revolution). So a 314.159 rad/s signal is at 50 Hz, an ω of 1 000 rad/s is at about 159 Hz, and an ω of 1 × 10⁶ rad/s is at about 159 kHz.

Angular frequency is preferred in physics because it keeps the trigonometry clean. The standard expression for a sinusoid in the time domain is x(t) = A sin(ωt + φ), with ω in rad/s. If you wrote it as x(t) = A sin(2π f t + φ) with f in Hz, the 2π would appear inside every derivative and every wave equation. Engineers and physicists pay the 2π conversion cost once, at the boundary between the spec sheet and the equation, so everything in the middle stays tidy. The cost is one multiply or divide by 2π — which is exactly what the frequency converter does.

Frequency and wavelength: the wave equation

For any wave, frequency and wavelength are tied together by the wave speed: c = f × λ, where c is the wave speed, f is frequency in Hz, and λ is wavelength in metres. For electromagnetic waves in vacuum (or close enough, in air), c = 299 792 458 m/s exactly — this is now a defined constant under the 2019 SI redefinition. So:

• 2.4 GHz Wi-Fi: λ = 299 792 458 / 2.4 × 10⁹ ≈ 0.125 m (12.5 cm). The familiar quarter-wave antenna at this band is about 3 cm long.
• FM radio (98 MHz): λ = 299 792 458 / 9.8 × 10⁷ ≈ 3.06 m. The classic dipole antenna for FM is a half-wavelength, about 1.5 m.
• 5G mid-band (3.5 GHz): λ ≈ 8.6 cm.
• 5G mmWave (28 GHz): λ ≈ 1.07 cm — short enough that obstacles like leaves and human bodies block the signal noticeably.
• Visible red light (430 THz): λ ≈ 697 nm.

For sound waves in air at 20 °C, c is about 343 m/s. So middle A (440 Hz) has a wavelength of about 0.78 m, the lowest note on a standard 88-key piano (A0, 27.5 Hz) has a wavelength of about 12.5 m, and the highest (C8, 4 186 Hz) is about 8.2 cm. The wavelength calculator handles the conversion either way and takes the wave speed as an input, so it works for light, sound, radio, and any other wave.

Reference frequencies to anchor a figure

Pure numbers across thirty orders of magnitude are hard to picture. A few reference frequencies make it easier to spot when a converted figure is obviously wrong:

• 1 μHz: roughly one cycle per 11.6 days — slower than tidal cycles.
• 1 mHz: one cycle per 1 000 seconds, about 17 minutes — typical of slow geophysical oscillations.
• 1 Hz: the human heartbeat is roughly 1 Hz at rest (60 bpm).
• 50/60 Hz: mains electricity.
• 440 Hz: concert A, the standard tuning pitch for orchestras.
• 20 Hz – 20 kHz: the conventional range of human hearing.
• 20 kHz – 200 kHz: ultrasound used in medical imaging and industrial cleaning.
• 540 – 1700 kHz: AM radio in most of the world.
• 88 – 108 MHz: FM radio.
• 433 MHz, 868 MHz, 915 MHz: license-free ISM bands used by LoRa, key fobs, smart-home devices.
• 2.4 GHz, 5 GHz, 6 GHz: Wi-Fi (and Bluetooth shares the 2.4 GHz band).
• 3 – 100 GHz: 5G mid-band and mmWave; satellite communications.
• 0.3 – 3 THz: the terahertz gap — between microwave and infrared light.
• 430 – 770 THz: visible light (red to violet).

If a converted figure puts an audible tone above 1 MHz or a Wi-Fi channel below 100 MHz, the conversion has gone wrong by at least three orders of magnitude, and it is worth re-checking. The frequency converter is exact, so a wildly off result almost always means the unit was misread or the prefix was dropped.

How to convert frequencies in your head

For mental estimation, a small set of shortcuts covers most common conversions:

• Hz → rpm: multiply by 60. So 50 Hz → 3 000 rpm; 100 Hz → 6 000 rpm.
• rpm → Hz: divide by 60. A 7 200 rpm hard drive is 120 Hz.
• Hz → rad/s: multiply by 6.283 (2π). So 50 Hz → 314 rad/s; 1 kHz → 6 283 rad/s.
• rad/s → Hz: divide by 6.283. So 1 000 rad/s → about 159 Hz.
• Hz to wavelength (electromagnetic, in metres): divide 300 by f in MHz. So 100 MHz → 3 m; 1 GHz → 0.3 m; 2.4 GHz → 0.125 m. Within 0.1% of the exact answer.
• Octaves: doubling the frequency raises a tone by an octave. So 220 Hz, 440 Hz, 880 Hz, 1 760 Hz are successive A notes.

These approximations are not meant to replace the converter — the exact answer is one input away — but they make it possible to spot when a quoted figure is wrong by an order of magnitude. If a spec sheet claims a 2.4 GHz antenna should be 1 m long, the 300 ÷ MHz shortcut tells you instantly that a quarter-wave at that band is closer to 3 cm.

Common mistakes

Dropping the SI prefix

The most common single mistake is reading "MHz" as "kHz" or the other way around, with a factor of 1 000 in the wrong direction. A "2.4 MHz" Wi-Fi signal would not work — that is an AM radio frequency. A "2.4 kHz" Wi-Fi signal is in the audible range and physically meaningless for a radio band. Always read the prefix twice, especially when copying from handwritten notes or scanned spec sheets where "k", "M", and "G" can blur into each other.

Forgetting the 2π for rad/s

Half of all rad/s mistakes are dropping the 2π entirely. The other half are multiplying by 2π when the conversion needs a divide (or vice versa). The reliable rule: rad/s is bigger than Hz by a factor of 2π, so going from rad/s to Hz divides, and going from Hz to rad/s multiplies. A 50 Hz mains supply has ω ≈ 314 rad/s, not 7.96 rad/s.

Treating angular velocity as frequency

A wheel spinning at 100 rad/s has an angular velocity of 100 rad/s, which corresponds to a frequency of about 15.9 Hz, which is about 955 rpm. All three numbers describe the same physical motion, but they are not interchangeable — using rad/s in a place that expects Hz produces an answer that is 2π too big. The frequency converter handles the bookkeeping; the responsibility is on the caller to feed it the unit the source actually used.

Mixing up mechanical rpm and electrical Hz

For a motor, the electrical mains frequency and the mechanical rotation frequency are linked by the pole count: n_rpm = (120 × f_Hz) / P. Converting 50 Hz to 3 000 rpm is only correct for a 2-pole motor. Most industrial motors are 4-pole (1 500 rpm at 50 Hz) or 6-pole (1 000 rpm at 50 Hz). Read the nameplate before assuming.

When the converter is not enough

For a real radio signal, the centre frequency is only part of the story. A modulated signal occupies a bandwidth around its carrier — a 2.4 GHz Wi-Fi channel is 20 MHz or 40 MHz wide depending on the standard, and a 5G mmWave channel can be 100 MHz or 400 MHz wide. The frequency converter converts the centre frequency but says nothing about the bandwidth or modulation; for link-budget or spectrum-planning work you need both numbers and the modulation scheme.

For music, the equal-tempered scale divides each octave into 12 semitones, with each semitone a factor of 2^(1/12) ≈ 1.0595 in frequency. So converting between "A" and "B♭" is not a unit conversion at all — it is a musical interval. The unit converter handles physical hertz; musical interval relationships need a separate tool.

For phase-noise or jitter analysis, the relevant quantity is not the frequency itself but the spectral density of the deviation away from it, measured in dBc/Hz or in ps RMS. The unit conversion is trivial; the measurement is not. Use a spectrum analyser or a dedicated phase-noise tool, not a units calculator.

Sources and standards

The exact SI prefix factors are from the BIPM SI Brochure, 9th edition (2019), which formalised the redefinition of the kilogram and fixed several physical constants exactly, including the speed of light at c = 299 792 458 m/s. The hertz is the SI derived unit of frequency, defined as one cycle per second (s⁻¹ in SI base units). NIST Special Publication 811 (2008, with updates) lists the recommended symbols and prefix conventions for US technical writing.

Radio spectrum allocations (Wi-Fi, mobile, broadcast, ISM bands) are set nationally by regulators — Ofcom in the UK, the FCC in the US, BNetzA in Germany — under the framework of the ITU-R Radio Regulations, which the World Radiocommunication Conference revises every three to four years. The frequencies quoted above are the typical international values; specific national allocations may differ at the edges. The frequency converter is unit-agnostic — the numbers it returns are correct under any allocation.

Related calculators

• Frequency Converter — the calculator this article supports.
• Wavelength Calculator — convert frequency to wavelength for any wave speed.
• Speed Converter — m/s, km/h, mph, knots, and other speed units.
• Time Converter — seconds, minutes, hours, days, years.
• Distance Converter — km, miles, feet, metres, nautical miles.