An astronaut is floating in free fall in deep space. His body mass, plus the mass of his spacesuit totals up to exactly 100kg. He has in his hand a 1-kg brick. As long as no forces are acting on the man, he will not move. His velocity and momentum are zero. This is a result of Newton's first law:

I. Every mass in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it.

If he throws the brick as hard as he can, the brick will travel away from him because he exerted a force on it with his hand. He will travel in the opposite direction, because the brick exerts a force on his hand. As long as the man and brick are in contact, force is being applied to each by the other, the moment they separate the force and accelleration stops. Both the man and the brick exerted forces on each other's mass, and a mass acted on by a force accellerates according to Newton's second law:

II. The relationship between an object's mass m, its acceleration a, and the applied force is F = ma.

Consequently, they will travel in opposite directions from the point in space where they where floating, i.e., there will be a recoil. That is Newton's third law

III. For every action there is an equal and opposite reaction.

However, the man is more massive than the brick, so it turns out that the man and the brick will travel away from the initial point in space at different speeds. Both the brick and man have stopped accelerating (no more forces involved) but each is now both moving at a constant velocity in opposite directions, and will continue to do so until another force intervenes.

Since the forces of man on brick and brick on man are equal (3rd law),

mass(man) x Accelleration(man) = mass(brick) x accelleration(brick)

by simple algebra, the brick's acceleration is 100 times the man's accelleration. It has to be to make up for the fact that the brick is 1/100 as massive as the man. However, man and brick were exerting forces on each other and accelerating each other for the same amount of time so if velocity = accelleration x time, then the velocity of the man must also be 1/100 the velocity of the brick, and in the opposite direction.

This relation will hold true no matter how hard the man throws the brick.

Also the mass(brick) x velocity(brick) = mass(man) x velocity(man) but in the opposite direction. Although they are different masses traveling at different speeds the mv of one cancels out the mv of the other since mv (MOMENTUM) is a vector quantity. The total momentum of the man- brick system remanins unchanged from when we started. It is 0. This is the Law of Conservation of Momentum, which always holds true. Knowing this can help you figure out other things because no matter what happens, this condition must be satisfied: the momentum before and after must remain unchanged. If your equations have an infinite number of solutions, only the one which satisfies this condition is the right one.

That's why the math matters.