Circled Plus

Usually XOR in computing, direct sum in algebra. XOR, short for exclusive or, gives true (1) when exactly one of two inputs is true: 1 ⊕ 0 = 1, but 1 ⊕ 1 = 0. On bits that is addition with the carry thrown away, which is why a plus inside a circle was chosen. You meet it in digital circuits, checksums, error-correcting codes and ciphers: the classic one-time pad encrypts by computing message ⊕ key.

In linear algebra and group theory, V ⊕ W is the direct sum: two structures placed side by side so that every element splits into one part from each, with no overlap. The plane is the direct sum of two lines, for example. A paper or textbook may also define ⊕ as some custom operation; when it does, its own definition wins.

Why is ⊕ next to an M in astronomy?

There ⊕ is the symbol for planet Earth. Astronomers write M⊕ for one Earth mass and R⊕ for one Earth radius, so an exoplanet of 5 M⊕ weighs five Earths. The circle crossed by a cross is an old astronomical sign for Earth; ♁ is its other form.

In organic chemistry diagrams, a plus in a circle marks an atom carrying a positive charge, and ⊖ marks a negative one.