2. Introduction
Number Systems, in mathematics, various
notational systems that have been or are
being used to represent the abstract
quantities called numbers. A number
system is defined by the base it uses, the
base being the number of different symbols
required by the system to represent any of
the infinite series of numbers. Thus, the
decimal system in universal use today
(except for computer application) requires ten
different symbols, or digits, to represent
numbers and is therefore a base-10 system.
3. Binary
(Base
2)
Octal
(Bas
e 8)
Decim
al
(Base
10)
Hexade
cimal
(Base
16)
0 0 0 0
1 1 1 1
10 2 2 2
11 3 3 3
100 4 4 4
101 5 5 5
110 6 6 6
111 7 7 7
1000 10 8 8
1001 11 9 9
1010 12 10 A
1011 13 11 B
1100 14 12 C
1101 15 13 D
1110 16 14 E
1111 17 15 F
1000
0
20 16 10
1111
1111
377 255 FF
1111
1010
001
372
1
2001 7D1
Number Systems
The decimal number system is in
universal use today, except in
computers, which uses the binary
number system. The decimal system
requires ten different symbols, or
digits, to represent all numbers; it is
known as a base-10 system. The
binary system requires only two
digits, 0 and 1. Some cultures have
used systems based on other
numbers. The Babylonians, for
4. Integer, any number that is a natural number (the counting
numbers 1, 2, 3, 4, ...), a negative of a natural number (-1, -2, -
3, -4, ...), or zero. A large proportion of mathematics has been
devoted to integers because of their immediate application to
real situations.
Any integer greater than 1 that is divisible only by itself and 1 is
called a prime number (see Number Theory). Every integer has
a unique set of prime factors, that is, a list of prime numbers that
when multiplied together produce the integer concerned. For
example, the prime factors of the integer 42 are 2, 3, and 7.
5. Rational Numbers, class of numbers that are
the result of dividing one integer by another.
Integers are the negative and positive whole
numbers (… -3, -2, -1, 0, 1, 2, 3, …). The
numbers ’, 5 (5/1), and –1.4 (-7/5) are
therefore rational numbers because they are
quotients (results of division) of two integers.
Rational numbers are a subset of the real
numbers, which also include the set of
irrational numbers.
6. Irrational numbers are numbers such as
pi (p), the square root of two (Ã), and the
mathematical constant e that are not the
quotient of any two integers.
All rational numbers can be written as
decimal numbers. The decimals may
have a definite termination point (such as
5 or 3.427) or they may endlessly repeat
in a pattern (such as 1.8888… or
2.18181…).
7. Irrational Numbers, class of numbers
that cannot be produced by dividing
any integer by another integer. Integers
comprise the positive whole numbers,
negative whole numbers, and zero: …-
3, -2, -1, 0, 1, 2, 3…. Examples of
irrational numbers include the square
root of two (Ã, 1.41421356…), pi (p,
3.14159265…), and the mathematical
constant e (2.71828182…).
8. When expressed as decimals these
numbers can never be fully written
out as they have an infinite number of
decimal places which never fall into a
repeating pattern.
The irrational numbers, together with
the rational numbers (numbers that
can be produced by dividing one
integer by another), make up the set
of real numbers.
9. Decimal numbers
Decimal numbers come from
fractions: They correspond to
particular fractions. Nowadays,
a decimal number indicates a
number written with a decimal
point and followed by a number
of figures, 23.45 for example
.
10. The idea of place values can be
extended to accommodate fractions.
Instead of writing 1‘(one and two-
tenths), we can use a decimal point
(.) to represent the same fraction as
1.2. Just as places to the left of the
decimal represent units, tens,
hundreds, and so on, those to the
right of the decimal represent.
11. Multiplying decimals is similar to multiplying
integers, except that the position of the
decimal point must be kept in mind. First,
multiply decimal numbers as if they were
integers, without considering the decimal
points. Then place the decimal point at the
appropriate position in the product so that
the number of decimal places is the same as
the total number of decimal places in the
numbers being multiplied. For example, in
multiplying 0.3 by 0.5
12. In cases where the divisor is a decimal number,
convert the problem to one in which the divisor is
an integer; division may then proceed as in the
above example. To divide 14 by 0.7, for example,
convert the divisor to an integer by multiplying it
by 10: (0.7)(10) = 7. Then multiply the dividend by
an equal amount. We can understand this
procedure more easily by considering the division
rewritten as a fraction. Multiplying both numerator
and denominator by the same amount will not
change the value of the fraction:
13. Negative numbers
There is evidence that negative numbers were being
used in India in the 7th century. It is important to
note that Indians used zero, a necessary
precondition to conceiving negative numbers.
Negative numbers were called debt numbers for
commercial reasons, just as today's bank statements
have two columns entitled credit (for receipts) and
debit (for expenditure).
14. The use of negative numbers in the West came much later. Italian
Renaissance mathematicians, who were specialists in algebra (the
science of solving equations), understood that without negative
numbers they could not solve certain equations (x + 7 = 0, for
example). However, they were unsure whether to call these
solutions proper numbers. And even in the 17th century, French
mathematician Descartes described negative numbers as false
numbers.
It was not until the 19th century that negative numbers were finally
treated as true numbers.
15. The arithmetic operation of addition is
basically a means of counting quickly
and is indicated by the plus sign (+). We
could place 4 apples and 5 more apples
in a row, then count them individually
from 1 to 9. Addition, however, makes it
possible to count all of the apples in a
single step (4 + 5 = 9).
We call the end result of addition the
sum. The simplest sums are usually
memorized. This table shows the sums
of any two numbers between zero and
nine:
16. To find the sum of any two numbers from 0 to
9, locate one of the numbers in the vertical
column on the left side of the table and the
other number in the horizontal row at the top.
The sum is the number in the body of the table
that lies at the intersection of the column and
row that have been selected. For example, 6 +
7 = 13.
We can easily add long lists of numbers with
more than one digit by repeatedly adding one
digit at a time. For example, if the numbers
27, 32, and 49 are listed in a column so that
all the units are in a line, all the tens are in a
line, and so on, finding their sum is relatively
simple:
17. 1. How do we decide the group(s) to which a number
belongs?
We can distinguish several groups of numbers.
denotes the set of whole numbers. = .
denotes the set of integers.
denotes the set of rational numbers. These are numbers that
can be written in the form where and * (therefore they
are whole quotients).
For example, and are rational numbers, but is not a
rational number.
denotes the set of real numbers. This set comprises all the
numbers that we use. They can be shown on a coordinate
number line:
18. Every real number is represented by a point and
each point represents a real number.
This set includes irrational numbers, i.e., real
numbers that are not rational.
These sets of numbers are subsets of each other
as follows: N,Z,R .
This means that a whole number is also an integer;
an integer is also a decimal, etc.
19. To recognize a number's type:
Simplify how it is written as much as possible.
In the case of an irreducible quotient a/b , carry out the
division. If it is finite (i.e., if the remainder is zero) then
a/b is a decimal; if it is infinite, it is referred to as
recurring and a/b is a rational number that is not a
decimal.
If the number cannot be written as a quotient of integers,
then it is irrational
20.
21. Pythagoras
Considered the first true mathematician, Pythagoras in the 6th century bc
emphasized the study of mathematics as a means to understanding all
relationships in the natural world. His followers, known as Pythagoreans, were the
first to teach that the Earth is a sphere revolving around the Sun. This detail
showing Pythagoras surrounded by his disciples comes from a fresco known as the
School of Athens (1510-1511), by Italian Renaissance painter Raphael.
22. Rules of calculation:
The sum of two positive numbers is positive;
the sum of two negative numbers is negative;
the absolute value of the sum of two numbers
with the same sign is the sum of the absolute
values of these numbers.
Examples:
(+7) + (+2) = +9. As (+7) and (+2) are positive,
the result is positive. We obtain 9 when we
calculate 7 + 2.
(-4) + (-6) = -10. As (-4) and (-6) are negative,
the result is negative. We obtain 10 when we
calculate 4 + 6.
B. The two numbers have different signs
23. Rules of calculation:
The sign of the sum of two numbers with
different signs is the sign of the number
with the greater absolute value;
the absolute value of the sum of two
numbers with different signs is the
difference between the absolute values of
the numbers (the largest minus the
smallest).
Particular case: The sum of two opposite
numbers is equal to 0. For example, (-
7) + (+7) = 0.
24. Examples:
(+9) + (- 4) = +5. With the numbers (+9) and
(-4), (+9) has the greater absolute value and
so it gives its + sign to the result. We obtain
5 when we calculate 9 - 4.
(+2) + (-8) = -6. In this second example, it is
(-8) that has the greater absolute value and
that gives its “–” sign to the result. We
obtain 6 when we calculate 8 - 2
