How can the number of reported decimals for a measurement tell about the instrument used in the measurement?

Question

How can the number of reported decimals for a measurement tell about the instrument used in the measurement?

and

What is the defnition of the slope of a line?

Answer 1

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Answer 2

All measurements include a margin of error, no matter how precisely we make the measurement, or how fine the design of  the instrument. The more the significant digits in the measurement, the lower the margin of error.

The slope of a line represents the change in the y coordinate divided by the corresponding change in the x coordinate. This is generally given as:

Δy/Δx = (y2 - y1)/(x2 - x1)

On a curved plot, the slope is the tangent to the curve at a given point and represents the instantaneous change in y at the point x. The smaller the Δx, the more accurate the slope represents the rate of change.

Answer 3

When measurements are taken in Physics (or any other science for that matter) they always have some degree of inaccuracy. The number of decimals, typically known as significant figures, shows how precise the measurement was taken. This is a lot easier to understand given an example:

Let's say that you had a meter stick. Now suppose that this stick has no markings on it besides a 1 meter mark. The best degree of accuracy you would be able to determine with this (to a reasonable precision) is entire meters. And then after you account for all your "full meters" you take a guess as to what is left over. So you might determine something to be 2.5 meters. Does this mean that the object is precisely 2.5 meters? Absolutely not: you were measuring with a ruler that only measured meters! The point is that the last digit is typically rounded and is an estimate or a "guess." With this meter stick could you could not, with any degree of accuracy, measure something that was exactly 2.49 meters in length because you don't have the precision required to do that.

So to answer your question the number of digits in a number shows the greatest accuracy that the instrument can measure, plus one additional number for rounding (guessing if you like). Adding any additional numbers to an instruments measurement is frivolous because they are meaningless to the calculation and only give the illusion of more accurate data.

For your second question the formal definition for the slope of a line is the change in the vertical (rise) divided by the change in the horizontal (run). The concept of slope is extensively used in calculus when the derivative is introduced to provide a mathematical way of determining instantaneous slopes along a curve.

Answer 4

the definition of the slope of a line is the  change in y/change in x.

the more decimals the more accurate the instrument has to be if the value is to have meaning. Think of measuring with a meter stick as compared to measuring with a micrometer.

Answer 5

This is a precision vs accuracy issue, the number of decimals (actually significant figures) implies the precision of the instrument, not the accuracy.  It is possible to be precisely inaccurate.  

Slope is what the others said, the change in X and Y coordinates, looks like a slope on a graph.