Dividing and subtracting same fundamental quantities.

 You are absolutely correct! When you numerically divide a length by another length, the dimension of length cancels out, resulting in a unitless quantity.

Here's why:

• Length is a fundamental physical quantity with its own dimension.
• Division involves calculating how many times one quantity fits into another.
• When dividing two lengths (e.g., meters by meters, centimeters by inches), you're essentially finding out how many times one length unit fits into another.

For example:

• Dividing 10 meters by 5 meters results in 2. In this case, we are finding out how many times 5 meters fit into 10 meters. The answer, 2, is a unitless quantity representing the number of times (2) the smaller length fits into the larger one.

The dimensions (meters) cancel out because we are not calculating a new length but rather a ratio or a scale between two lengths.

This concept applies to other fundamental quantities as well. Dividing mass by mass results in a unitless quantity representing a ratio of masses. Dividing time by time results in a unitless quantity representing a ratio of times (e.g., how many times one second fits into one hour).

You're right, subtracting a fundamental quantity from another fundamental quantity generally doesn't result in the units canceling out. Here's why there's a difference between subtraction and division:

Subtraction:

• Subtraction implies finding the difference between two quantities of the same kind.
• For example, subtracting 5 meters from 10 meters makes sense because they are both lengths. The result, 5 meters, remains a length because you're finding the difference in distance between two points.

Division:

• Division involves calculating how many times one quantity (of a certain kind) fits into another quantity (of the same kind).
• As explained earlier, dividing two lengths (e.g., meters by meters) results in a unitless ratio representing how many times one length unit fits into another.

Key Points:

• Subtraction applies to quantities of the same kind and maintains the unit.
• Division calculates a ratio between quantities of the same kind and often results in a unitless quantity.

Exceptions to Subtraction Rule:

There can be some exceptions where subtracting quantities of the same fundamental kind might lead to a different unit. Here are a few examples:

1. Temperature Scales: Subtracting temperatures in Celsius or Fahrenheit from each other makes sense (assuming they measure the same starting and ending points). However, the resulting unit remains °C or °F, respectively, because these scales are based on relative differences from a reference point, not absolute temperatures.

2. Derivatives: In calculus, when finding the derivative of a position function (meters with respect to time), the result is velocity (meters per second). Here, the unit changes because the derivative represents the rate of change of the position, which involves both distance and time.

Overall, it's important to understand the context and the kind of fundamental quantities involved in subtraction. In most cases, subtracting them will maintain the unit, but there can be a few exceptions depending on the specific scenario.