A mathematics textbook chapter introducing integers, their representation on a number line, and basic addition rules.
This is a mathematics textbook chapter titled 'Numbers' that introduces the concept of integers. It is written for students beginning a course in basic number theory or arithmetic. The chapter starts by defining positive integers as counting numbers 1, 2, 3, 4 and explains the need for zero as a number to represent the absence of count.
The text describes natural numbers as the set of positive integers together with zero. It shows how natural numbers can be represented geometrically on a line using a unit of distance, with the point 0 called the origin. The chapter also introduces negative integers, written as -1, -2, -3, to represent temperatures below zero or years before Christ.
The chapter explains that positive integers, negative integers, and zero together are called integers. It discusses applications of integers for measuring distance, temperature, and time, using the birth of Christ as the origin for years. The text then moves to addition rules, starting with adding a positive number by moving right on the number line.
The section on addition covers adding negative numbers by moving left on the number line. It provides examples such as 5 + (-3) = 2 and 10 - 15 = -5. The text introduces the concept of additive inverse, stating that a + (-a) = 0 and -a + a = 0, and shows that a and -a lie on opposite sides of zero.
The chapter concludes with the rules for addition: commutativity (a + b = b + a) and associativity ((a + b) + c = a + (b + c)). Examples include 3 + 5 = 5 + 3 and (-2) + 5 = 5 + (-2). The text notes that the sum and product of integers are also integers, setting up the next sections on multiplication rules.