Mathematics starts with natural human languages, such as Bablonian, Egyptian, Greek, Mayan, Sanskrit, Arabic, or English. The language is used to describe math, and the math tools are used to construct the models which help us describe and understand the world and universe. This is used in fields such as all the sciences, all the engineering, social sciences such as economics, as well as business, cooking, clothes making, construction, food production, all forms of entertainment, and everything related to computers.

The first connection to natural human languages are words which are used. They are given specific meaning, sometimes multiple meanings, but usually only a few meanings. English has some words, like "set" and "run", with hundreds of meanings. Mathematics tries to avoid this. In this blog, some effort will be made to reduce the "overloading" of key words, finding other words to take on some of the meanings. In math, two other tactics are used to reduce the words used, as well as their meanings: using many more symbols, and splitting math for different math fields as well as different application fields.

Some of the words include

• Definitions: Statements of what words mean. These are usually widely accepted, though there are some situations where other options may be worth considering. These will often come from other fields.
• Axioms, postulates, or properties: Statements which are widely accepted as true, often without proof. These are core building blocks for all of mathematics, so the goal is to have as few as possible. If any statement which is assumed to be true turns out to be untrue, chunks of mathematics can collapse.
• Theorems: Statements which are proven. These are as reliable as the underlying assumptions and the logical steps used. An important aspect is that proofs can generally be followed or reproduced by anyone with the needed foundational understanding. There is no "gatekeeping" in mathematics, everyone is welcome to, and encouraged to, participate.
• Notation: The symbols used in mathematics.

When ideas such as definitions, axioms, theorems, notations, and a few others, are introduced, an attempt will be made to designate them with the appropriate label, starting now.

• [Definition] operator: An action which connects items in a precise way.
• [Definition] operand: The item or items acted upon by an operator.

Addition

• [Definition] addition: Combining items in a specific way.
• [Notation] \(+\) : The operator used to indicate addition.
• [Definition] evaluate: To use operators and operands to get a single quantity.
• [Definition] result: The single quantity obtained when using operators and operands.
• [Definition} sum (total): The result of the addition operation.
• [Notation] \(=\) (equals) : The symbol used to connect the operation to its result. This is one of the most overloaded notations.
• [Notation] \(\rightarrow\) (equals, the result is): An option instead of "\(=\)" with a more precise use.
• [Definition] number: One of the core items used in mathematics, starting with counting but building up from there.
• [Definition] algebra: A core field of mathematics dealing with quantities having unspecified values. It is one of the most important fields of mathematics, used in most others. Some fields will introduce their own key ideas, and then use algebra to develop the field.
• [Definition] geometry: Another "core" field of mathematics, which is somewhat distinct from algebra, but significantly benefits from an algebraic approach.
• [Definition] numerical algebra: a sub-field of algebra which concentrates on numerical quantities.
• [Definition] arithmetic: The part of numerical algebra which only uses numbers and the core operators. Some say that arithmetic is not a subset of algebra, but I have yet to hear a good argument for why, since everything which gets presented in arithmetic applies in algebra. By treating arithmetic as a component of numerical algebra, it will be natural to quickly move from arithmetic to algebra, and from there, to other fields of mathematics.

Those are a lot of words, now it's time to use them in a mathematical way: \begin{align} 5 + 3 &= 8 \\ 5 + 3 &\rightarrow 8 \end{align}The numbers, \(5\) and \(3\), are added together to give the result, \(8\)

• [Axiom] commutative property of addition: With the addition operator, the order of the operands does not matter:
  \begin{align} 5 + 3 &= 3 + 5 = 8 \\ 5 + 3 &== 3 + 5 \rightarrow 8\end{align} If we have a pile of pebbles, for example, we can count out five, and then three more, getting eight. We can then count out three, and then five more, and again get eight, the same number of pebbles in each pile.
• [Definition] equality: States or tests if two quantities have the same value, if they "are equal to" each other.
• [Notation] \(==\) (equality): To further reduce the overloading of \(=\), the double equal sign is used to indicate equality. This comes from the computer science world.

Generalizing into "algebra mode", replace the numbers with identifiers,
\begin{align} a + b &== b + a\end{align} While this prevents the evaluation to get a result, it does allow the first step in algebraic manipulation.

• [Definition] algebraic manipulation: Moving quantities in mathematically valid ways to allow simplification to isolate a particular quantity, or to emphasize some aspect of what is being manipulated. Presenting techniques to do this will be one of the core goals of this blog.

Multiplication

• [Definition] multiplication: An operation to represent repeated addition.
• [Definition] product: the result of a multiplication operation.
• [Notation] \(\times\): one of the operators used to represent the multiplication operation, which can also be written algebraically: \begin{align} 5 + 5 + 5 &= 5 \times 3\ =\ 15 \\ 5 \times 3 &\rightarrow 15 \\ a \times b &\rightarrow c \end{align} The numbers, \(5\) and \(3\), are multiplied together to give the result, \(15\). While this is demonstrated using simple numbers like \(5\) and \(3\), the equivalent algebraic form suggests that much more can be done.
• [Axiom] commutative property of multiplication: As with the addition operator, the order of the operands for the multiplication operator does not matter: \begin{align} 5 + 5 + 5 &= 3 + 3 + 3 + 3 + 3 = 15 \\ 5 \times 3 &== 3 \times 5 \ \rightarrow 15 \\ a \times b &== b \times a \end{align}