Scalars

PhysicsMath
A scalar is a quantity that is the same in all inertial frames.

It has a magnitude but it lacks a direction.

A scalar can change the type of an object. As an example, mass changes acceleration to force.		A scalar is a math object.

A scalar is something that multiplies to change the component of something but it doesn’t change the “type” of that something.

Examples

PhysicsMath
\overrightarrow{F} = m \overrightarrow{a}

F is a vector with units of force.
a is a vector with units of displacement per time squared.

m is a scalar with units of mass and as such it is capable of multiplying against the units of a to produce the units of F.		2 \begin{pmatrix} 3 \\ 4 \end{pmatrix} = \begin{pmatrix} 6 \\ 8 \end{pmatrix}

If you haven’t seen matrices before, you can probably see what the 2 does to the \begin{pmatrix} 3 \\ 4 \end{pmatrix}
Comparison of a Physicist’s Scalar to a Mathematician’s Scalar

If there was a war between the physicists and the mathematicians over scalars, the mathematicians would probably win.

When we come to Vector Spaces, the word scalar is used in the “mathematician” way.

Later in the program I will argue that we like what the math people do, so much, that we will encourage them to “go crazy” and work to see things differently. They make stuff that we like.

We need for you to see a scalar as what physicists see because equations like…

\overrightarrow{a} = b \overrightarrow{c}

…where b is a scalar that has units. It is not a number; it is a number with “baggage”.

ChatGPT said it well:

The physical units associated with scalar quantities are essential … (they) provide a link to the underlying physical reality.