## F‑numbers and the aperture have an inverse relationship

Hi there, my name is Paul, and this is Exposure Therapy. In this video, I’ll explain the reason for the inverse numerical relationship between f‑numbers and the aperture. This relationship is a widespread point of confusion for many beginner photographers, who regard it as irrational or needlessly complex. My goal is to dispel the mystery around f‑numbers and demonstrate why they’re a perfectly reasonable method for expressing how the aperture affects exposure.

Understanding the relationship between picture brightness and both the shutter speed and ISO is straightforward for students learning the basics of photography. Shutter speed is expressed numerically in time units, with the most common being fractions of a second; longer durations result in brighter pictures, and shorter durations result in darker pictures. ISO is also expressed numerically; bigger numbers produce brighter photos, and smaller numbers make darker photos.

In both cases, the relationship between the setting and its effect on picture brightness is easy to understand because there’s a positive correlation, and they move in tandem. For example, when you double the exposure duration, it doubles the brightness; when you halve the ISO, it halves the brightness. It’s a simple relationship that students in my photography workshops grasp with ease.

## The inverse relationship between f‑numbers and the aperture is confusing for many beginners

Unfortunately, the relationship between f‑numbers, aperture size, and picture brightness is not as immediately intuitive. Beginners are confused by the negative (or inverse) relationship between f‑numbers and aperture size. In addition, they have a hard time understanding why bigger f‑numbers represent smaller apertures that reduce brightness, and smaller f‑numbers define larger apertures that increase brightness.

The best way to address this is by starting with the basics. Inside every interchangeable lens is a ring of overlapping blades collectively known as an iris diaphragm or iris. Expanding or contracting the blades adjusts the opening in the centre of the iris, called the aperture.

## Introducing the Entrance Pupil

When you hold a lens up and look at the aperture, what you’re seeing is technically called the “entrance pupil.” The entrance pupil is the optical image of the physical aperture as seen through the front of the lens. This distinction matters because when you look at the front of a lens, you see the aperture through multiple layers of glass that affect its magnification and perceived location in space compared to the physical opening in the iris. For the sake of simplicity, I’ll use “aperture” when referring to both the setting and the physical opening and “entrance pupil” in reference to dimensions.

Changing the size of the aperture adjusts the intensity of light passing through the lens. Increasing the aperture’s size allows more light to pass through the lens, increasing exposure and creating a brighter picture. Conversely, decreasing the aperture’s size reduces how much light passes through the lens, reducing exposure and resulting in a darker photo.

## Why are apertures expressed using f‑numbers?

We express aperture values using f‑numbers and not as the measured size of the entrance pupil, such as its diameter, radius, or area, because it neglects the essential role of focal length. This can be demonstrated with a thought exercise.

Let’s pretend we have two lenses attached to identical cameras: one lens is 50 mm and the other is 100 mm, and both have entrance pupils with 25 mm diameters. Since their entrance pupils are identical in size, an equal amount of light enters each lens. However, because the focal length of the 100 mm lens is twice that of the 50 mm lens, the light passing through it has to travel twice the distance to reach its camera’s image sensor, which produces a darker image.

Reduction in brightness occurs because light has the property of spreading out as it recedes from its source, and from the perspective of your camera’s image sensor, this source is the point inside the lens from which focal length is measured. This trait of light to diffuse outwards is described by the Inverse Square Law, which states that intensity is inversely proportional to the square of the distance. In this example, the inverse square law informs us that the 100 mm lens exposes its camera’s image sensor to 1/4 the light compared to the 50 mm lens because it’s twice as long. This occurs because one over two squared equals one-quarter.

The 100 mm lens can provide an exposure equal to its 50 mm counterpart by opening its aperture to collect four times more light, assuming its aperture can open that much. Since apertures are roughly circular, we can determine how big they should be by calculating the area of a circle. An entrance pupil with a 25 mm diameter has an area of about 491 mm^2. The 100 mm lens would need an entrance pupil with an area of 1,964 mm^2, which is formed by a circle with a 50 mm diameter. Simple, right?

## F‑numbers express ratios

Fortunately, photographers don’t need to perform such calculations to take pictures! That’s because hidden within these numbers is a straightforward relationship. For example, notice how the exposure produced by the 50 mm lens with a 25 mm entrance pupil is identical to the 100 mm lens with a 50 mm entrance pupil. This is because in both cases, the ratio of the focal length to the entrance pupil diameter is 2:1.

This is precisely why the f‑number is sometimes called the f‑ratio. The f‑number expresses a ratio of the lens focal length to the diameter of the entrance pupil, and it’s defined by the equation N=ƒ/D. Thus, the f‑number equals the focal length divided by the entrance pupil diameter. It can also be modified to solve for the entrance pupil diameter using the equation D=ƒ/N. Thus, the entrance pupil diameter equals the focal length divided by the f‑number.

These equations demonstrate that choosing the same f‑number on a lens of any focal length will result in the same amount of light passing through the lens. They also explain the inverse relationship between f‑numbers and exposure. For a given focal length, as the aperture’s size increases, the ratio decreases, and vice versa.

A 50 mm lens set to ƒ/4 will have an entrance pupil diameter of 12.5 mm—because 50 divided by 12.5 equals 4. A 24 mm lens set to ƒ/8 will have an entrance pupil diameter of 3 mm. Some lenses can open to ƒ1.0, in which case the entrance pupil diameter and focal length are equal.

## The f‑number scale

The standard f‑number scale is: 1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22, 32, and so on. The difference in exposure between adjacent numbers is one stop, which means that it either doubles or halves the amount of light passing through the lens depending on whether you’re opening or closing the aperture. However, the numeric sequence grows by a factor of about 1.4 or shrinks by a factor of about 0.7.

Most photographers simply commit the standard f‑number scale to memory. However, if you’re having trouble, a more straightforward method is to remember just the first two numbers—1 and 1.4—because the rest of the scale is an iteration of doubling each in alternating order. The next f‑number is always double the previous one. So the number after ƒ/1.4 is double of ƒ/1, which is ƒ2. Likewise, the number after ƒ/2 is double of ƒ/1.4, which is ƒ/2.8. And on and on it goes.

Lastly, doubling the f‑number, such as changing it from ƒ/2.8 to ƒ/5.6, reduces picture brightness by one-quarter. And conversely, halving the f‑number, such as adjusting from ƒ/8 to ƒ/4, increases picture brightness four times.

I hope this helped you understand the inverse numerical relationship between f‑numbers and their effect on the aperture. If you have requests for future topics, let me know in the comments, and I’ll address them in future videos. In the meantime, you can learn more about photography on ExposureTherapy.ca. See you next time.