Metric

A metric is a function that defines distance between any two points in a set.

A metric makes all distances between adjacent integers the same.

A metric makes a truth like 6-4=7-5 possible.

Equal separation of integers results in equal distance between adjacent integers.

You’ve been working with math for so many years that it’s easy for your brain to think, how could these things not be true? That’s a fair question and we have one scenario below to help you with your escape.

Consider the exits on the turnpike. The distance between Exit 5 and Exit 4 is probably not the same as the distance between Exit 6 and Exit 5. Therefore 5 – 4 does not equal 6 – 5. You are allowed to do tricks like 1) measuring travel time driving between exits (constant speed) or 2) using the trip odometer, to verify the inequality of distance.

Without some external frame of reference, we couldn’t verify that the distances were unequal.

Defining the Metric

If there is a metric, then the following is true:

  • Distance between points ‘a’ and ‘b’ can be calculated by function, and for here that function will be \: d(a,b).
  • d(a,b) \geq 0
  • d(a,b) = 0 \iff a=b
  • d(a,c) \leq d(a,b) + d(b,c)

The last equation is known as the Triangle Inequality.

Appendix A

The metric might be the most abstract math concept that you have encountered.

For an idea like this, we recommend that if you find different explanations, you write them all down together to allow comparison.

We also ask you, what other ideas are strange and bizarre. If you are one of the students at Mockingbird Academy, be bold and ask us to pay you for writing reports on these things.

Appendix B

Occasionally in math we find statements where they say ‘a’ implies ‘b’. If you have spent some time dealing with logic your brain will immediately scream the question, does ‘b’ imply ‘a’?

At this point your brain is demanding that the speaker put truth into one of the two forms below – – it can’t be both:

We can use our geometric intuition to believe that a zero distance means the two points are the same. We can also use our geometric intuition to believe that if two points are the same, the distance is zero.

  • a implies b
  • a iff b

To cut to the chase, Encyclopedia Britannica online came out and said that the statement d(a,b) implies a=b is actually “if and only if (iff)” so b implies a is also true.

From our previous discussions we have said that a new idea believed can end up in two places. In the first scenario what we believe is converted into an axiom. In the second scenario what we believe is packed into a definition.

We threw a caution flag on this though. In math is always been about precise precision. If everybody writes if a then B, might there be a reason why they wrote it this way? We know we can’t totally rely on our geometric intuition to guess the correct answer for everything that there is in math. We know that our geometric intuition is biased toward the euclidean versions of everything.