NUMBER THEORY
Number theory is that branch of mathematics devoted primarily to the study of numbers especially integers.
DEFINITION of a number: A number is an idea. It is something you cannot see
but can only think about.You only think about a number when you affiliate it to
something. You see. For example 2 pens, 3 pencils, 5 bags etc. Five therefore
is an idea of collecting objects together.
In our system of counting called Arabic or Decimal system of
counting, we represent number base on the idea of groupings in tens. We use
digits to represent numbers as below:
Zero, one, two, three, four, five, six, seven, eight, nine.
0 , 1 , 2 , 3
, 4 , 5 , 6 , 7 ,
8 , 9 .
All other numbers are obtained from the above digits by combining
two or more of them and some times by comparing two of them. In each case the value
(place value) of a digit in the number depends on its position in the number.
CLASSIFICATION
OF NUMBERS
1) COUNTING NUMBERS. A counting number is a
positive whole number. Example 1, 2, 3,4,The set of counting numbers is written
as IN. The set of counting numbers is written as IN*, Hence IN*={
1,2,3,4…} The three dots in the set indicate that the set is so large and all
its members cannot be listed.
2) NATURAL NUMBERS. A natural number is a
counting number but also includes the number zero (0). It is a positive whole
number starting with zero. The set of natural numbers is denoted by IN and
written as IN= ﻉ0, 1, 2, 4…}
3) INTEGER NUMBERS. An integer number is
either a positive whole or negative whole number including zero. For example
0,3,5,6,... etc. The set of integers is denoted by Z and written as Z=} -2, -1, 0, 1, 2…}. So, Z the set of integers can be split into
three parts namely
i-the set of
positive integers denoted Z+ and written as Z+=ﻉ1,2,3,…}
ii-the set of
negative integers denoted Z- and
written as Z-=ﻉ…-3,-2,-1}
iii-the neutral set or empty set .
Hence Z= Z++Z-+
neutral or empty set.
4) RATIONAL NUMBERS. A number is said to be rational if
it can be expressed as a fraction. That is , it can be expressed in the for ﻉp/q, q≠0 , p,q ﻉZ.}.A
rational number is denoted by Q. From the here 4 is a rational number because
it be written as 4/1.That means all integers are rational numbers. Hence, Zﮯ Q where
the symbol, ﮯ represents subset of.
5) IRRATIONAL NUMBERS. Any number which is not rational
is irrational. An irrational number cannot be expressed as a fraction. For
example Ï€, √2, 3√3 etc. The set of irrational numbers is denoted by
Qc.
6) REAL NUMBERS. A real number is any number which is not
Complex. It is counting number, natural number, integer number, rational number
or irrational number. The set of rational numbers is denoted by IR. So
IR={…-3,-2,-1,-1/2,0,1/2,1,2,3,…}. Observe that here all the sets above are
subsets of the set of IR.
7) DIRECTED NUMBERS. There are two signs on any number
which indicate the direction of the number on the number line. These signs are
positive (+) and negative (-). Zero is neutral; it is neither positive nor
negative. Any number without a sign is considered positive unless it is zero.
On the number line, negative numbers are on the left of zero while positive
numbers are on the right .
As you move from left to
right of the number line ,numbers increase . That is to say -3>-4, -2>-3,
0>-1, 1>0, 2>1, 3>2, 4>3 and so on. As you move from right to
left the numbers decrease that is 3<4, 2<3,1<2,0<1, -1<0,
-2<-1,-3<-2,-4<-3 etc. The arrow heads pointing to the left and right
shows that there are many more numbers in these directions.
8) ABSOLUTE VALUES
OF NUMBERS. The absolute value of a positive number remains positive while that
of a negative number becomes positive. In general, the absolute value of a
number,x, denoted by ↾x↾. It means ↾x↿ is the positive value ox. E.g. I5I=5,
I-5I=5
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