Math Questions - Why does TMM consider zero a natural number? Why does TMM define 0^0 as equal to 1?

There are two disputable matters in mathematics on which Leigh has chosen to take a stand.

• First, we include 0 as a natural number.
• Second, we define 0^0 as equal to 1.

Because these are disputable matters, we are very clear in our definition of each one. We recognize that not everyone agrees with us (often vociferously!), but that is the nature of a disputable matter. Students in the upper challenges, and especially in the Transcendental domain, will have the opportunity to explore this idea of mathematical disputable matters and how the field in which they are working may define these terms differently.

There is no firm consensus in the math world about zero. Some include it with the naturals, while others do not. One reason that we chose to include zero with the natural numbers was to allow us to define the digits as a subset of the naturals. This mathematical inconsistency is a great reminder of the importance of defining our terms.

Whole and counting numbers are terms used in K-12 education but not really used in number theory - we had as our goal to use the most precise math terms that we could, so we chose to go with the number sets most recognized in mathematical studies.

We do define the counting and whole numbers, but we work with naturals, integers, rationals, irrationals, reals, and complex as the number sets by which we sort numbers. We discuss transcendental vs algebraic, but that is really only pertinent for our oldest students.

Zero may or may not be included in the natural numbers depending on who you read or talk to. We chose to include zero in the naturals. You may encounter other curricula or texts that do not include zero in natural numbers, but it isn’t a firmly established fact in mathematics.

I once read that poetry is using lots of words to say the same thing and mathematics is using 1 word to say many things. :)

The symbol | | does indeed mean absolute value. When we consider the meaning of absolute value, it is easier to see why the same symbol is used to represent the cardinality of sets.

Absolute Value, cardinality, and modulus are all terms represented by | | and all of them mean magnitude. A magnitude is always positive and represents size or distance. Thus, cardinality is the magnitude of a set measured in the number of elements. Absolute value is the distance of a single number from the origin |-3| = |3| or the distance between two numbers, |3 - 5| The modulus of a complex number is the distance of the complex number from the origin |3 + 4i| and is calculated using the Pythagorean theorem.

In the Naturals Beta/Digital lessons, students are introduced to these ideas separately and as grammar work. As they become familiar, they will be able to synthesize the ideas as I did above.