I am wondering if there is a basic unit in math. 3 <mtext>&#xA0;</mtext> c m &#x00

Let's take a different tack with this. Imagine learning arithmetic the way it was in the past. The need for counting is the first issue you run into. As we go hunting, you notice a herd of cattle (what did people hunt in the past?). You should also let the hunting party know how many cattle are present when you return. In this case, you say a number, like n, and tell the group that it's cattle, because n. Therefore the head of cattle would be the unit here.
This is an interesting problem for a mathematician. We had to have come up with a counting system that is systematic throughout all units. For example, 1 cattle and 1 apple have the same count, but they have different units. As pure mathematics grew, the units became tools to relate these numbers. Essentially, we could use the natural numbers, $\mathbb{N}$, to express the count of objects in a set. This is referred to as the cardinality of a set.
Later, people realized that these counting numbers could also represent lengths. But another problem arose: how do you express a number in the middle of 0 to 1 or 1 to 2? In this sense we expanded the definition of the number system to include the positive rational numbers, ${\mathbb{Q}}^{\mathbb{+}}$. Further units were added to give these numbers meaning as mathematics developed and more people applied it to the world.

This is advantageous in the real world.

As you have noted, it is very normal to recognize the connection between units and numbers. Understanding the meaning behind math and mathematical things is a pretty natural thought.

Nevertheless, numbers go beyond the bounds of our common sense.

Numbers are unitless objects in the strictest sense.

Mathematical objects are "ideas" that behave exactly like the number "1" in every way that we can think of.

whatever its size or shape.
As we expanded our set of numbers to include $\mathbb{C} , \mathbb{H} , \mathbb{O}$it becomes difficult to relate these numbers to a universal system of measurement because what meaning does it make to say that we have $1 + i$ cattle? It makes very little sense at all! If you are interested in this subject however, this (as I see it) is a good way to view the creation of set theory. Also, if you want to learn more about this type of thing, mathematics is the right place to look!

The number 1 is the "basic unit of measurement". It is a scalar multiple of all other numbers.