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No, weight is not a fundamental quantity. Though mass is.

The reason for adopting this viewpoint is this: the mass of a body is intrinsic to the body itself - e.g. an electron has the same mass anywhere in the universe. The weight of a body is just the gravitational force acting on that body, and obviously depends on the gravitational field it finds itself in.

For example, your mass remains the same if you fly to the moon, but your weight on the moon's surface is roughly one-sixth of your weight here on Earth.

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