Mathematics Fundamentals — Things

As we begin an examination of my vision of twenty-first century mathematics, as it should be, we must establish some fundamentals. Let’s get started. This is important stuff.
One of the things we do in mathematics is use deductive reasoning to study relations and operations on mathematical objects. As you read mathematics it will help your understanding if you identify each “thing” as either:

  • a relation
  • an operation
  • a mathematical object

Try to make that identification as soon as you are introduced to each new mathematical “thing”.
Here are some examples to help you understand what I am talking about.
Each of the following is a mathematical object.

  • numbers
  • variables
  • algebraic expressions
  • formulas
  • geometric figures

Each of the following is a binary relation.

  • Equal to
  • Less than
  • Greater than
  • Equivalent to

Each of the following is a binary operation.

  • Addition of Real Numbers
  • Subtraction of Real Numbers
  • Multiplication of Real Numbers
  • Division of Real Numbers
  • Composition of functions

Each of the following is a unary operation.

  • Forming the opposite of a number
  • Forming the reciprocal of a number
  • Calculating the square root of a non-negative real number
  • Calculating the absolute value of an algebraic expression
  • Raising a real number to the fifth power

You will be introduced to additional objects, relations, and operations as you continue reading mathematics.

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