Frequency counts how many cycles happen per second. Like waves hitting a beach or your heart beating. Measured in hertz (Hz). f = 1/T where T is period. Higher Hz = faster oscillation.

• 1 Hz = 1 cycle per second
• Frequency = 1 / period (f = 1/T)
• Higher frequency = shorter period
• Fundamental for waves, oscillations, rotation

Frequency and period are reciprocals. f = 1/T, T = 1/f. High frequency = short period. 1 kHz = 0.001 s period. 60 Hz AC = 16.7 ms period. Inverse relationship!

• Period T = time per cycle (seconds)
• Frequency f = cycles per time (Hz)
• f × T = 1 (always)
• 60 Hz → T = 16.7 ms

For waves: λ = c/f (wavelength = speed/frequency). Light: c = 299,792,458 m/s. 100 MHz = 3 m wavelength. Higher frequency = shorter wavelength. Inverse relationship.

• λ = c / f (wave equation)
• Light: c = 299,792,458 m/s exact
• Radio: λ in meters to km
• Light: λ in nanometers

Hz is SI unit (cycles/second). Named after Heinrich Hertz. Prefixes from nano to exa: nHz to EHz. 27 orders of magnitude! Universal for all oscillations.

• 1 Hz = 1 cycle/second
• kHz (10³), MHz (10⁶), GHz (10⁹)
• THz (10¹²), PHz (10¹⁵), EHz (10¹⁸)
• nHz, µHz, mHz for slow phenomena

Angular frequency ω = 2πf (radians/second). RPM for rotation (revolutions/minute). 60 RPM = 1 Hz. Degrees/time for astronomy. Different perspectives, same concept.

• ω = 2πf (angular frequency)
• RPM: revolutions per minute
• 60 RPM = 1 Hz = 1 RPS
• °/s for slow rotations

Radio engineers use wavelength. f = c/λ. 300 MHz = 1 m wavelength. Infrared: micrometers. Visible: nanometers. X-ray: angstroms. Frequency or wavelength—two sides of same coin!

• Radio: meters to km
• Microwave: cm to mm
• Infrared: µm (micrometers)
• Visible/UV: nm (nanometers)

f = 1/T (frequency from period). ω = 2πf (angular frequency). λ = c/f (wavelength). Three fundamental relationships. Know any quantity, find others.

• f = 1/T (period T in seconds)
• ω = 2πf (ω in rad/s)
• λ = c/f (c = wave speed)
• Energy: E = hf (Planck's law)

All waves obey v = fλ (speed = frequency × wavelength). Light: c = fλ. Sound: 343 m/s = fλ. Higher f → shorter λ for same speed. Fundamental wave equation.

• v = f × λ (wave equation)
• Light: c = 3×10⁸ m/s
• Sound: 343 m/s (air, 20°C)
• Water waves, seismic waves—same law

Photon energy: E = hf (Planck constant h = 6.626×10⁻³⁴ J·s). Higher frequency = more energy. X-rays more energetic than radio. Color = frequency in visible spectrum.

• E = hf (photon energy)
• h = 6.626×10⁻³⁴ J·s
• X-ray: high f, high E
• Radio: low f, low E

AM radio: 530-1700 kHz. FM: 88-108 MHz. TV: 54-700 MHz. WiFi: 2.4/5 GHz. 5G: 24-100 GHz. Each band optimized for range, bandwidth, penetration.

• AM: 530-1700 kHz (long range)
• FM: 88-108 MHz (high quality)
• WiFi: 2.4, 5 GHz
• 5G: 24-100 GHz (high speed)

Visible: 430-750 THz (red to violet). Infrared: <430 THz (thermal, fiber optics). UV: >750 THz. X-rays: EHz range. Different frequencies = different properties, applications.

• Red: ~430 THz (700 nm)
• Green: ~540 THz (555 nm)
• Violet: ~750 THz (400 nm)
• Infrared: thermal, fiber (1.55 µm)

Human hearing: 20-20,000 Hz. Musical A4: 440 Hz. Audio sampling: 44.1 kHz (CD), 48 kHz (video). Video: 24-120 fps. Heart rate: 60-100 BPM = 1-1.67 Hz.

• Audio: 20 Hz - 20 kHz
• A4 note: 440 Hz
• CD audio: 44.1 kHz sampling
• Video: 24-120 fps

Each prefix = ×1000. kHz → MHz ÷1000. MHz → kHz ×1000. Quick: 5 MHz = 5000 kHz.

• kHz × 1000 = Hz
• MHz ÷ 1000 = kHz
• GHz × 1000 = MHz
• Each step: ×1000 or ÷1000

f = 1/T, T = 1/f. Reciprocals. 1 kHz → T = 1 ms. 60 Hz → T = 16.7 ms. Inverse relationship!

• f = 1/T (Hz = 1/seconds)
• T = 1/f (seconds = 1/Hz)
• 1 kHz → 1 ms period
• 60 Hz → 16.7 ms

λ = c/f. Light: c = 3×10⁸ m/s. 100 MHz → λ = 3 m. 1 GHz → 30 cm. Quick mental math!

• λ = 300/f(MHz) in meters
• 100 MHz = 3 m
• 1 GHz = 30 cm
• 10 GHz = 3 cm

FM station at 100 MHz. What's the wavelength?

λ = c/f = (3×10⁸)/(100×10⁶) = 3 meters. Good for antennas!

Motor spins at 1800 RPM. Frequency?

f = RPM/60 = 1800/60 = 30 Hz. Period T = 1/30 = 33.3 ms per revolution.

Light at 600 nm wavelength. What frequency and color?

f = c/λ = (3×10⁸)/(600×10⁻⁹) = 500 THz = 0.5 PHz. Color: orange!

Concert pitch (A above middle C) standardized at 440 Hz in 1939. Before that, varied from 415-466 Hz! Baroque music used 415 Hz. Modern orchestras sometimes use 442-444 Hz for 'brighter' sound.

Human eye most sensitive to 555 nm (540 THz) green light. Why? Sun's peak output is green! Evolution optimized our vision for sunlight. Night vision peaks at 507 nm (different receptor cells).

Frequency chosen because water molecules resonate near this frequency (actually 22 GHz, but 2.45 works well and penetrates deeper). Also, 2.45 GHz was unlicensed ISM band. Same band as WiFi—can interfere!

Electromagnetic spectrum spans 30+ orders of magnitude. Visible light (400-700 nm) is less than one octave! If EM spectrum were piano keyboard spanning 90 keys, visible light would be single key.

Modern CPUs run at 3-5 GHz. At 5 GHz, period is 0.2 nanoseconds! Light travels only 6 cm in one clock cycle. This is why chip traces matter—signal delay from speed of light becomes significant.

Highest-energy gamma rays from cosmic sources exceed 10²¹ Hz (zettahertz). Photon energy >1 MeV. Can create matter-antimatter pairs from pure energy (E=mc²). Physics gets weird at these frequencies!

Heinrich Hertz proves electromagnetic waves exist. Demonstrates radio waves. Unit 'hertz' named after him in 1930.

IEC adopts 'hertz' as unit of frequency, replacing 'cycles per second'. Honors Hertz's work. 1 Hz = 1 cycle/s.

A4 = 440 Hz adopted as international concert pitch standard. Previous standards varied 415-466 Hz.

Hertz officially adopted in SI system. Becomes standard for all frequency measurements worldwide.

Meter redefined from speed of light. c = 299,792,458 m/s exact. Ties wavelength to frequency precisely.

CPU frequencies reach GHz range. Pentium 4 hits 3.8 GHz (2005). Clock speed race begins.

SI redefinition: second now defined by cesium-133 hyperfine transition (9,192,631,770 Hz). Most precise unit!

Hz measures cycles per second. RPM measures revolutions per minute. They're related: 60 RPM = 1 Hz. RPM is 60× larger than Hz. Motor at 1800 RPM = 30 Hz. Use RPM for mechanical rotation, Hz for electrical/wave phenomena.

One complete cycle = 2π radians (360°). If f cycles per second, then ω = 2πf radians per second. Example: 1 Hz = 6.28 rad/s. Factor of 2π converts cycles to radians. Used in physics, control systems, signal processing.

Use λ = c/f where c is wave speed. For light/radio: c = 299,792,458 m/s (exact). Quick: λ(m) ≈ 300/f(MHz). Example: 100 MHz → 3 m wavelength. Higher frequency → shorter wavelength. Inverse relationship.

Chosen because water absorbs well near this frequency (water resonance actually at 22 GHz, but 2.45 penetrates better). Also, 2.45 GHz is unlicensed ISM band—no license needed. Same band as WiFi/Bluetooth (can interfere). Works well for heating food!

Visible spectrum: 430-750 THz (terahertz) or 0.43-0.75 PHz (petahertz). Red ~430 THz (700 nm), green ~540 THz (555 nm), violet ~750 THz (400 nm). Use THz or PHz for light frequencies, nm for wavelengths. Tiny slice of EM spectrum!

Mathematically, yes (indicates phase/direction). Physically, no—frequency counts cycles, always positive. In Fourier analysis, negative frequencies represent complex conjugates. In practice, use positive values. Period also always positive: T = 1/f.