Square Root of '2'

introduction

This number comes to 1.414, to the nearest three decimal places, but simpler if just thought of as 1.4 instead.  It may seem obscure but the value of √2 is used in several places which you may not be aware of.

Printing dot angles

FIGURE 1

Black dots in a printed image are set to 45° to give the best dot pattern. This means that the printed dots are at 45° to the image pixels from which the dots have been created. Since the pixels remain in a horizontal and vertical pattern, the black printer dots are now on a diagonal that is √2 times longer. As the diagonal length has increased 1.4 fold, then the resolution of the pixels supplying information to the dots is only 0.71 times this (the inverse of 1.4).  Therefore the pixel resolution needs to increase by 1.4 to hold enough information for creating the black printer dots.

Camera F stops

Camera lens F Stops use an apparently odd set of numbers. The problem is that the amount of light entering the lens is governed by the area of the aperture; a squared function, not a linear one. If the F Stops increased in a 2 fold linear form (1,2,4,8...) then the exposure would not increase two fold but as a square of this, which is 4 fold.  Opening the aperture by one F Stop would therefore quadruple the exposure as 2² = 4. So a set of numbers is needed, that when squared comes to 2, to give the double exposure required. In other words, numbers need to increase by √2, as √2² = 2, to give twice the area.
•  1  x √2 = 1.4
• 1.4 x √2 = 2
• 2  x √2 = 2.8
• 2.8 x √2 = 4
•  4  x √2 = 5.6
• 5.6 x √2 = 8  etc.
Hence the familiar sequence of 1, 1.4, 2, 2.8, etc, used on camera lenses. This now means that one F Stop increase doubles the light exposure, and likewise one F Stop decrease halves it.

Metric paper sizes

Metric A paper sizes are designed to be twice the area of each smaller A size. The A0 is twice the area of A1, which in turn is twice the area of A2, etc. Notice that it is twice the area, not twice the size. As the square root of 2 is 1.4, then multiplying the size measurements of an A4 by 1.4 would give you an A3. As both the width and length are multiplied by this, then the paper area is doubled as √2² = 2.
• A5 x √2 = A4
• A4 x √2 = A3
• A3 x √2 = A2
• A2 x √2  = A1
• A1 x √2 = A0
The inverse of 1.4 = 0.71, so multiplying an A3 by 0.71 results in an A4. This is worth remembering if you use a photocopier without A paper size settings, as an A3 sheet set to 71% will result in a A4 photocopy.