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Long before the architects of ancient Egypt, the astronomers of Babylon, or the ancient Greek mathematicians; the simple farmers needing to trade their crops or animals in the market was the impetus that created the finite positive integers. However, zero was not "discovered" until sometime between 400 to 300 BCE!

There are many ways to write a numerical quantity with the most common expression being as a decimal number. However, binary, octal, and hexadecimal representations of numbers are also frequently used, e.g., in computer science. How can a number, which in the language of math is often conveyed with the word "scalar," possibly be unique if there are so many different ways to convey its value? Do scalars even need to be unique? If yes, then must that uniqueness be absolute, true in all mathematics; or, will a scalar being relatively unique suffice, meaning only unique within some math topics? Obviously, both situations simultaneously existing cannot be true.

Math Fact OO:

The sum of two odd integers is always an even integer.

Math Fact EE:

The sum of two even integers is always an even integer.

Math Fact EO (or OE):

The sum of an even integer and an odd integer

(or the sum of an odd integer and an even integer)

is always an odd integer.

EXAMPLE: The Importance of Uniqueness

Starting with the usual decimal representation of numbers, consider the sum 11+ 3 = 14. This sum is just one of very many instances characterized by Math Fact OO.

Let us change the base used for the numerical expressions (where the meaning of the word "base" is similar to how many letters are in a linguistic alphabet) from ten to three. After this change the translation of the above sum has become: 102 + 10 = 112. However now, this second sum's equation looks a lot more like an instance from Math Fact EE; and, not at all like it is an instance belonging to Math Fact OO.

So what happened? Are the Math Facts OO and EE no longer valid? Worse, are the above Math Facts only be valid in some bases, such as base-10; while not valid in others, such as base-3? If you think this difference is a result of going from an even base to an odd base; then consider that the sum's base-5 translation is 21 + 3 = 24: in support of Math Fact OO.

Thus, In the absence of any clarifying formats, it appears the base-3 sum, written as 102 + 10 = 112, does not support Math Fact OO; yet, more importantly this base-3 sum does not contradict Math Fact OO.

Moreover, it also superficially appears the decimal sum, 11 + 3 = 14, does not provide any support for Math Fact EE; however, once again do notice the more important subtlety that this base-10 sum does not contradict Math Fact EE.

Aside Remark 1: Math Fact EO (or OE) was only provided here for the sake of completeness regarding integer addition because every integer is either even or odd, but is never both even and odd.

Therefore to avoid any confusion regarding the base used in an arithmetic expression, it is customary for a base that is not equal to ten to have its value written as a subscript following every term or symbol in the arithmetic expression. As such, in the base-3 sum the subscript of 3 (written "_{3}") follows the last digit of each number, the plus sign and the equals sign, which is the displayed expression

102_{3} +_{3} 10_{3} =_{3] 112_{3}

However, the above arithmetic looks pretty messy! Avoiding this very visually cumbersome "_{3}" notation is the foremost reason machine generated mathematics is typeset, simply because the typeset process employs a subscript sized font positioned lower on the line of text.

Aside Remark 2: In computer science any octal number, base-8, is designated as such with the leading character "0", i.e., an extraneous zero in front of the octal number. All hexadecimal numbers, base-16, are written with the leading characters "0x". Given the following decimal numbers 12, 15 and 124, their respective octal and hexadecimal representations are: 014; 017; 0174; and, 0xC; 0xF; 0x7C.

Recall, hexadecimal numbers, i.e., base-16, use the conventions:

the decimal value of 10 is written as an A or a;

the decimal value of 11 is written as a B or b;

the decimal value of 12 is written as a C or c;

the decimal value of 13 is written as a D or d;

the decimal value of 14 is written as an E or e;

the decimal value of 15 is written as an F or f.

In summary, regardless of the language used to relate information between people; humans, as a rule, want the original message and its translation to convey the same meaning. Uniqueness provides this important benefit to the math of number theory, which is an ongoing conversation that is communicated with the language of mathematics.

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