Arithmetic -1
Dear student, Today I'll discuss about The Arithmetic in details. Read attentively this following post.
Properties of Integers
An
integer is any number in the set
{. . . −3, −2, −1, 0, 1, 2,
3, . . .}. If
x and
y are integers and
x ≠
0, x is a
divisor (factor) of
y provided that
y =
xn for some integer
n. In this case
y is also said
to be
divisible by
x or to be a
multiple of
x. For example, 7 is a divisor
or factor of 28 since
28 = 7 × 4, but 8 is not a divisor of 28
since there is no integer
n such that
28 =
8n.
Integer
Quotients and remainders
If x and y are
positive integers, there exist unique integers q and
r, called the quotient and
remainder, respectively, such that y
= xq + r and 0 ≤ r <
x. For example, when 28 is divided by 8, the
quotient is 3 and the remainder is 4 since 28 = (8)(3) + 4. Note
that y is divisible by x
if and only if the remainder r is 0; for example, 32
has a remainder of 0 when divided by 8 since 32 is divisible by 8. Also note
that when a smaller integer is divided by a larger integer, the quotient is 0
and the remainder is the smaller integer. For example, 5 divided by 7 has the
quotient 0 and the remainder 5 since 5 = (7)(0) + 5.
Odd and even integers
Any integer that is divisible by 2 is an even integer; the set of
even integers is {. . . −4, −2, 0, 2, 4, 6, 8, . . .}. Integers
that are not divisible by 2 are odd integers; {. . . −3, −1, 1,
3, 5, . . .} is the set of odd integers.
If at least one factor of a product of integers is even, the product is even;
otherwise the product is odd. If two integers are both even or both odd, their
sum and their difference are even. Otherwise, their sum and their difference are
odd.
Prime numbers
A prime number is a positive integer that has exactly two different
positive divisors, 1 and itself. For example, 2, 3, 5, 7, 11, and 13 are prime
numbers, but 15 is not, since 15 has four different positive divisors, 1, 3, 5,
and 15. The number 1 is not a prime number, since it has only one positive
divisor. Every integer greater than 1 is either prime or can be uniquely
expressed as a product of prime factors. For example, 14 = (2)(7), 81 =
(3)(3)(3)(3), and 484 = (2)(2)(11)(11).
Consecutive integers
The numbers −2, −1, 0, 1, 2, 3, 4, 5 are consecutive integers.
Consecutive integers can be represented by n,
n + 1, n + 2, n +
3, . . ., where n is an integer. The numbers
0, 2, 4, 6, 8 are consecutive even integers, and 1, 3, 5, 7, 9 are
consecutive odd integers. Consecutive even integers can be represented
by 2n, 2n +
2, 2n + 4, . . ., and
consecutive odd integers can be represented by 2n + 1, 2n +
3, 2n + 5, . . ., where n is an integer.
Properties of the integers 1 and 0
If n is any number, then 
, and for
any number 
. The number 1 can be expressed in many ways, for example, 
for any number 
.
Multiplying or dividing an expression by 1, in any form, does not change the
value of that expression.
The integer 0 is neither positive nor negative. If n is any number, then n
+ 0 = n and 
.
Division by 0 is not defined.
By:
Mohi Uddin
On 7:05 AM
,
n is the numerator and d is the denominator.
The denominator of a fraction can never be 0, because division by 0 is not
defined.
and
are equivalent since they both represent the number 
.
and 
.
and 
, multiply the numerator and denominator of the first fraction by 7
and the numerator and the denominator of the second fraction by 5, obtaining
and 
, respectively; 
.
, the lcm of 3 and 6 is 6 (not 
), so
.
.
.
is 
. In general, the reciprocal of a fraction 
is 
, where 
, is a mixed number. 
means 
.
.
