Historically, mathematics came into being to serve two purposes, counting and measuring. Both of these required the use of numbers, the positive integers ℕ and the real numbers ℝ, respectively. The need to solve equations such as 2x=3, x+4=0, x2+1=0 subsequently led to the appearance of more sophisticated number systems like the rational numbers ℚ, the integers ℤ, and the complex numbers ℂ. We are chiefly concerned here with properties of the positive integers and, at the same time, the means by which such properties are established. This revolves around the concept of a mathematical proof,of which we give examples of four kinds, finishing up with the most important for ℕ, proof by induction.
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