Dimensional Formula of Frequency: [M⁰ L⁰ T⁻¹] and Hertz

The dimensional formula of frequency is [M⁰ L⁰ T⁻¹], and with the zero powers dropped that is simply [T⁻¹]. Frequency counts how many complete cycles happen in one second, and a count has no dimensions, so only the "per second" survives.

Its SI unit, the hertz, is therefore just s⁻¹. A tuning fork marked 256 Hz vibrates 256 times every second; a mains supply at 50 Hz reverses direction 100 times a second, completing 50 full cycles. For the period, wave and oscillator formulas themselves, open the frequency formula page.

Quick facts

Dimensional formula
[M⁰ L⁰ T⁻¹], short form [T⁻¹]
Defining formula
$f = \frac{1}{T}$
SI unit
hertz (Hz), 1 Hz = 1 s⁻¹
Other forms
$f = \frac{v}{\lambda}$ , $f = \frac{\omega}{2\pi}$
Type
Scalar
Common symbols
f or ν (Greek nu)

Building the dimensional formula of frequency from a cycle count

Frequency is the number of cycles n in a time t: $f = \frac{n}{t}$ .
A number of cycles is a pure count: $[n] = [M^{0}L^{0}T^{0}]$ .
So $[f] = \frac{1}{[T]} = [T^{-1}]$ .
For one cycle, n = 1 and t is the period T, giving the familiar $f = 1/T$ .

The wave equation gives the same answer. From $v = f\lambda$ , $f = \frac{v}{\lambda} = \frac{[LT^{-1}]}{[L]} = [T^{-1}]$ . Showing both routes in a long answer is a good habit, because they test each other.

From [T⁻¹] to the hertz, and some conversions

Replacing T with the second turns the bracket into s⁻¹, which SI names the hertz. Larger multiples are common: kHz for sound, MHz for radio, GHz for wireless networks and processors.

Example: a radio station broadcasts at 98.3 MHz. Find its period and its wavelength in air.

Given: $f = 98.3 \times 10^{6}\ \text{Hz}$ , $c = 3.00 \times 10^{8}\ \text{m/s}$ .

Period: $T = \frac{1}{f} = \frac{1}{9.83 \times 10^{7}} = 1.02 \times 10^{-8}\ \text{s}$ .

Wavelength: $\lambda = \frac{c}{f} = \frac{3.00 \times 10^{8}}{9.83 \times 10^{7}} = 3.05\ \text{m}$ .

Revolutions per minute is also a frequency. Divide by 60 to get hertz: 1200 rpm is 20 Hz.

Checking an oscillation formula

Example: is $f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$ correct for a block on a spring?

Spring constant: $[k] = [MT^{-2}]$ . Mass: [M].

$\frac{[k]}{[m]} = [T^{-2}]$ , and its square root is $[T^{-1}]$ . The 1/2π is a number. The formula is dimensionally correct.

Example: a student writes the pendulum frequency as $f = \frac{1}{2\pi}\sqrt{\frac{l}{g}}$ . Spot the error.

$\frac{[L]}{[LT^{-2}]} = [T^{2}]$ , so the root is [T], a time. That is the period form upside down. The frequency must be $\frac{1}{2\pi}\sqrt{\frac{g}{l}}$ .

Quantities with the same bracket as frequency

Things measured per second
Quantity	Typical symbol	Usual unit
Frequency	f, ν	Hz
Angular velocity	ω	rad/s
Angular frequency	ω	radian per second
Velocity gradient	dv/dx	s⁻¹
Radioactive decay constant	λ	s⁻¹

Since these share [T⁻¹], dimensional analysis cannot separate them, and it cannot detect a missing 2π. Look at the physics: hertz is used for cycles, rad/s for angles turned. Also compare with a product such as hf, Planck's constant times frequency, which gives energy, as you can confirm from the bracket of h.

Mistakes when writing frequency dimensions

Writing [T]. That is the period. Frequency is its inverse.
Giving cycles a dimension. A cycle is a count, so it has none.
Confusing ν with v. The Greek letter nu for frequency looks like v for velocity in handwriting. Write it carefully.
Equating ω and f numerically. They share a bracket but differ by a factor of 2π.

For quick look ups of [T⁻¹] and its neighbors, the dimensional formulas page groups every oscillation and wave quantity.

Frequently asked questions

What is the dimensional formula of frequency?

It is [M⁰ L⁰ T⁻¹], written [T⁻¹]. Frequency is a count of cycles divided by time, and a count has no dimensions.

What is the dimension formula of frequency in terms of period?

Since f = 1/T, its bracket is the inverse of the period's bracket. Period is [T], so frequency is [T⁻¹].

Why is the hertz equal to s⁻¹?

The hertz means one cycle per second. The cycle is a pure number, so the unit reduces to one over a second.

Do frequency and angular frequency have the same dimensions?

They do: each is [T⁻¹]. Angular frequency ω equals 2πf, and 2π has no dimensions, so only the numbers differ.

Is frequency a vector quantity?

No, frequency is a scalar. It tells you how often something repeats, not which way anything points.

How are the brackets of frequency and time period related?

They are exact inverses. The time period of one oscillation has [M⁰ L⁰ T¹], and frequency, being 1/T, flips the time power to minus one. Multiply a frequency by its period and every dimension cancels to leave the pure number 1.