The real numbers we are so accustomed to (ℝ) extends from -∞ to +∞. However, it is overlooked that real numbers subtly have 2 components:
- a natural (ℕ) number component {0, …, ∞}; and
- a direction (the positivity or negativity of the absolute number).
Numbers vs Quantities
There is a subtle but profound difference between a quantity and number. If we understand this difference, it will quickly distil the nature of vectors and all subsequent concepts.
Technically, a quantity is a sensation of there being a simultaneous existence of an identifiable quality separated by another quality. Suppose you see 2 apples on a table. You identify that there is a repeating pattern of the quality of “appleness” separated by another quality, distance. The quality that repeats and the quality that separates can vary depending on what we are counting. You identify 26 letters in the alphabet because you identify the quality of “letterness” (the lines that make up a letter) and the distinct changes in the lines. Perhaps you identify 2 types of vehicle transmission: automatic and manual. The repeating pattern is the quality of gear changing, separated by the quality of mechanism. The nature of quantity can be philosophically intimate, but understanding the nature of quantity at this level will clarify the path we are taking.
A number is nothing more than an algebraic term that consists of the evaluation of an expression. We might see the number 3, but it can consist of multiple quantities, say \(1 + 2\) or \(-2 + 3 + 2\). All are the same number.
This philosophical approach is an unorthodox teaching. It is unique to this tutorial. Most mathematicians are not aware of this distinction explicitly but they know it intuitively after experience.
For example, -5 has a natural number of 5 in the negative direction. 3 has a natural number of 3 in the positive direction. The positivity is implied. It’s only by custom we omit the + symbol.
Suppose you have \($3\) in your bank account and you wish to represent it as a number. You could simply write \(3\) of course. In linear algebra, this is a scalar. It is a mere quantity with no aspect of dimension. We wish to represent our account as a vector, therefore we express it as:
\(\begin{bmatrix} 3 \end{bmatrix}
\\ \)
This is a vector. This new notation seems redundant, and it is. That’s why we don’t typically write number as vectors. We typically don’t care that a number is only one dimension, we just care for the quantity. When a number consists of a single quantity, we omit the notation.
Now suppose you have 2 accounts and thus wish to represent a 2 dimensional number, one quantity with $3 and another quantity with $2. You wish to represent it as a single number. You might think to simply add the numbers together, however this would result in a loss of information. We wouldn’t know how much each account has separately. Therefore, we write it as:
\(\begin{bmatrix} 3 \\ 2 \end{bmatrix}
\\ \)
This is a 2-dimensional vector. We can think of it as a single number and we can treat it as a single number algebraically, even assigning it to a variable.
\(a = \begin{bmatrix} 3 \\ 2 \end{bmatrix}
\\ \)
Now suppose you have your 2-dimensional vector and you wish to double the amount in both accounts. Normally we would write separate expressions \(3 \times 2\) and \(2 \times 2\). With vector notation, we can represent this in one single expression.
\(\begin{array}{}
\begin{bmatrix} 3 \\ 2 \end{bmatrix} \times 2 & = &
\begin{bmatrix} 6 \\ 4 \end{bmatrix}
\end{array}
\\ \)
It may seem tedious at this point to invent a new notation simply to represent 2 operations. It is. However, as data and the operations become more complex, the value of such notation becomes very useful.
To be continued.