Units in Science

Units in Science

All variables in physics (like energy E) are symbols that represent a number plus some units.

Variable = "number + units"

Ultimately this is because all quantities in science are define directly by a measuring process (operationally definition) or through an equation using variables that are themselves measurable (derived quantities). See Role of Measurement in Science.

Without units there is no way to know what standard the value of a variable is being compared. Even if you know that the variable represents some quantity like distance, x = 12.6 is meaningless unless you also specify the standard that the number is being compared - is it 12.6 m, 12.6 cm, 12.6 km, 12.6 mm, 12.6 light years, etc.
Unless otherwise specified, the variable you see in any equation in physics refer to establish set of units. Currently this is SI set of Units.

Stated more strongly, unless you put numbers into an equation that are expressed in SI units then you will get an answer but it will not represent a number that is associated with the SI units of quantity you calculated. For example, the SI units of velocity is meter/second, that of mass is kilograms, and that of energy is joules. If you place numbers whose units are in km/hr and grams into the equation for kinetic energy, KE = 1/2 m v², you will get an number but it will not be in joules. The numerical answer obtained will represent an energy but not in any form that I have ever seen used.

Rules involving Units in Equation

Let "a" and "b" represent two physical quantities or - more generally, mathematical expressions of physical variables.

a = b	The units of a and b must be the same if the equation is a valid equation.
a + b or a - B	The units of a and b must be the same if the expression is to be valid.
sin(a) tan(a) cos(a)	The unit of a have to be either radians or degrees. (If a is in radians then your calculator must be set to radian mode or the value you get will not be correct - like wise for degrees, you need to set your calculator into degree mode.) Actually radians and degrees are really not true unit in physics but represent place a maker specifying how a circle is divided up.
ln(a) e^a	The quantity a represents has to be unitless - it can not have any units at all.
a^. b or a/b	The quantities that a and b represent could have any units. The units of the expression (a^.b or a/b) is equal to the product or division of the units of a and b.