This document discusses number theory and properties of integers. It introduces Carl Gauss' statement about number theory being the "queen of mathematics". Key properties discussed include closure laws for integers under addition and multiplication, the absolute value of integers, and representation of integers as positive or negative. Summation notation is also introduced to simplify writing sums and products of integers.

1. NUMBER THEORY
Carl Friedrich Gauss, a great mathematician, once
remarked that “mathematics is the queen of sciences, but
number theory is the queen of mathematics”. Number
theory is the simplest of all types or branches of
mathematics that even those without much mathematical
training find it very interesting.
Properties of Integers
The set of integers (denoted by Z)
Z {..., -3, -2, -1, 0, 1, 2, 3, ...}
plays a significant development of the concept of number.
It possessed properties that developed mathematical ideas
and expounded salient facts.

2. The theory of numbers is primarily concerned
with the properties of the natural numbers
1, 2, 3, ..., also called the counting numbers or
positive integers. However, the theory is not
confined strictly to the set of natural numbers of
the set of integers.
It is supposed that the student understands
(from earlier math subjects) the following
properties which the set of integers obey.
1. Closure Laws
For any integers a and b, a + b Z and
a · b Z. However, Z is not closed with respect to
division.

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