Arithmetic is the oldest branch of mathematics dealing with basic operations like addition, subtraction, multiplication and division. It includes counting, as well as more advanced calculations used in science and business. The fundamental arithmetic operations are performed on different types of numbers, and concepts like order of operations, fractions, decimals, ratios, exponents, and roots are part of this subject. Key ideas in arithmetic also include factors and multiples, prime and composite numbers, the greatest common factor, lowest common multiple, and operations on fractions like finding a common denominator.

1. Basic Arithmetic
Arithmetic is the oldest branch of
mathematics, used by almost everyone. Its tasks
range from the simple act of counting to
advanced science and business calculations. The
traditional arithmetic operations
are addition, subtraction, multiplication, and divi
sion, although more advanced operations such
as percentages, square
root, exponentiation, and logarithmic
functions are also a part of this subject.

2. Types of Numbers
• Real number
• Rational number
• Integer
• Natural number
• Irrational number
• Odd number
• Even number
• Positive number
• Negative number
• Prime number
• Complex Number

3. Arithmetic operations and related concepts
• Order of operations
• Addition (+)
– Sum
• Subtraction (-)
• Multiplication (. /*/×)
– Multiples
• Common multiples
– Least common multiple (LCM)
• Division (÷)
– Quotient
– Fraction
• Decimal fraction
• Proper fraction
• Improper fraction
• Ratio
• Common denominator
– Lowest common denominator
– Factoring
• Fundamental theorem of arithmetic
• Prime number
• Composite number
• Factor
– Common factors
» Greatest common factor (GCF)
• Power
– Exponent
• Square root
• Cube root

4. Lowest Common Multiple (LCM)
 The LCM is something that you will use throughout math. It is especially useful when
multiplying and dividing fractions.
 When finding an LCM, use only multipliers that are whole numbers. Examples: 4, 8, 43, 104
Be sure to be aware of all the numbers you are finding an LCM for.
Problem: Find the LCM of 4 and 5.
Solution: Multiples of 4
4
8
12
16
20
...
Multiples of 5
5
10
15
20
25
...
Common Multiples 20…
20 is a multiple of both numbers.
It is also the first one (lowest of all multiples), thereby being the lowest common multiple.

5. Greatest Common Factor (GCF)
When finding GCFs, be aware of all the numbers you are finding common factors of and
remember that you can only use whole numbers for factors. When finding a GCF, unlike
the LCM, you must list all the factors because you're finding a greatest factor, not a
lowest multiple.
Problem: Find the GCF of 8 and 12.
Solution: Factors of 8
1
2
4
8
Factors of 12
1
2
3
4
6
12
Common Factors
1
2
4
4 is a factor of both numbers.
It is the largest of the factors listed, therefore it is the greatest common factor.

6. Common Denominator
When finding a common denominator so you can add or subtract fractions, you find the LCM
of all denominators of the fractions you are dealing with. Once you've found this
number, make the denominators equal this number. To do this, you multiply the
denominator and numerator (the denominator is one factor of the LCM) by the
corresponding factor of the LCM.
Problem: 4 2
-- + --
3 5
Explanation
Solution:
15
The LCM of 3 and 5 is 15.
4 2
----- + -----
3*5 5*3
Since the denominators have to equal the LCM,
you have to multiply 3 by 5 and 5 by
3. Now both denominators are the same.
4*5 2*3
----- + -----
15 15
Because you don't want to change the
problem in any way, each part of the problem has to
be multiplied by 1 (not one-third or one-fifth as
you did in the second step). To do that, you have to
multiply the numerator by the same number as
you multiplied the denominator by.
20 6
---- + ----
15 15
Now that you've got the denominators the same,
you can add the fractions together.
26
----
15
You cannot reduce this fraction,
so this is the final answer.

7. Factoring
Factor: x2 - 14x – 15
Solution: First, write down two sets of
parentheses to indicate the
product. ( )( ) Since the first term in
the trinomial is the product of the
first terms of the binomials, you
enter x as the first term of each
binomial. (x )(x ) The product of the
last terms of the binomials must
equal -15, and their sum must
equal -14, and one of the
binomials' terms has to be
negative. Four different pairs of
factors have a product that equals -
15. (3)(-5) = -15 (-15)(1) = -15 (-
3)(5) = -15 (15)(-1) = -15 However,
only one of those pairs has a sum
of -14. (-15) + (1) = -14 Therefore,
the second terms in the binomial
are -15 and 1 because these are
the only two factors whose product
is -15 (the last term of the
trinomial) and whose sum is -14
(the coefficient of the middle term
in the trinomial). (x - 15)(x + 1) is
the answer.

8. Remember the Binomial formulae?
• In elementary algebra, the binomial theorem describes the
algebraic expansion of power of a binomial. According to the
theorem, it is possible to expand the power (x+y)^n into a
sum involving terms of the form ax^by^c, where the
exponents b and c are nonnegative integers with b+c=n, and
the coefficient a of each term is specific positive integer
depending on n and b. When an exponent is zero, the
corresponding power is usually omitted from the term.
The formal expression:
where

9. e.g.
• (a+b)4 = b4 + 4ab3 + 6a2b2 + 4a3b +a4
The binomial coefficients
can be given a meaning even if p is not a positive integer, so long as k is positive
integer.
We put
and
For
This conforms to the case where p is a positive integer since
For example

10. Factorials
• Factorials are very simple things. They're just
products, indicated by an exclamation mark. For
instance, "four factorial" is written as "4!" and
means 1×2×3×4 = 24. In general, n! ("enn factorial")
means the product of all the whole numbers
from 1 to n; that is, n! = 1×2×3×...×n.
• 0! is defined to be equal to 1, not 0. Memorize this
now: 0! = 1

11. Recall that the factorial notation "n!" means " the
product of all the whole numbers
between 1 and n", so, for instance, 6! =
1×2×3×4×5×6. Then the notation "10C7" (often
pronounced as "ten, choose seven") means:

12. Theory of Sets
• A set is a collection of objects called elements of the set.
• To discuss and manipulate sets we need a short list of symbols commonly
used in print. We start with five symbols summarized in the following
table.

13. Venn Diagram
• To visualize relationships between 2, 3 or 4 sets we quite often use
pictures called Venn diagrams. The standard way to draw these diagrams
for 1, 2 and 3 sets is illustrated here. The enclosing rectangle represents
the universe.