From Wikipedia:
The SI base units and their physical quantities are the metre for measurement of length, the kilogram for mass, the second for time, the ampere for electric current, the kelvin for temperature, the candela for luminous intensity, and the mole for amount of substance.
As far as I know, a base is a unit which cannot be broken down into units other than those from the above mentioned units. However, degrees (for angles) and 8-bit bytes (for digital data) cannot be expressed using one or more of the base units. So, why are these two units not considered base units?
Answer
The radian (not the degree) is the SI unit of angle, and it's defined in terms of lengths: it is that angle for which the length of a circular arc subtending that angle is equal to the radius of the circle. Since this definition refers to the relative ratio of two lengths, the SI considers it to be a "dimensionless derived unit", rather than a base unit.1
As far as bytes go: Defining a unit amounts to specifying a certain amount of a quantity that we call "one unit". Physical quantities such as mass, length, time, etc., are (effectively) continuous quantities, and so there is no "natural" unit for us to use. We therefore have to make an arbitrary choice about how much of each quantity is equal to one unit.
Digital information, on the other hand, is inherently discrete. All methods of quantifying data simply amount to counting bits; and you don't need to make an arbitrary choice of unit if you can simply count a quantity. There is therefore no need to define a unit for digital information, because there is already a natural unit (the bit).
It's important to note that not every measurable quantity is inherently definable in terms of SI base units. If I count the number of people in my office building right now, and tell you that there are "12 people" in the building right now, then "people" is not expressible in terms of meters, kilograms, and seconds. But I don't need to worry that you're going to come along and use some different unit to count the people in this building, because a natural unit (1 person) exists. It's only when we are measuring a quantity that can take on any real-numbered value (e.g., the mass of all the people in this building) that it becomes important to define a unit; otherwise, you and I have no basis for comparison. Any system of units is essentially a set of these arbitrary choices; "natural" units of quantities that are inherently discrete are unnecessary simply because they're understood to be the obvious choice.
1 It's worth noting that the radian was officially a "supplementary unit" in SI until 1995, when they were reclassified as "dimensionless derived units". A bit of the discussion surrounding this change can be found on p. 210 of the Proceedings of the 20th Conférence Générale des Poids et Mesures (warning: large PDF). Reading between the lines, I suspect that the name "dimensionless derived unit" was something of a compromise between those who thought it should be thought of as a derived unit and those who didn't think it should be thought of as a unit at all; but I wouldn't want to speculate further than that.
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