Multiplication & Division of Integers

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Multiplication of whole numbers is the mathematical operation of adding together multiple copies of the same number. For example, four multiplied by three is twelve, since three sets of four make twelve:

$4 + 4 + 4 = 12.\!\,$

Multiplication can also be viewed as counting objects arranged in a rectangle, or finding the area of rectangle whose sides have given lengths.

Multiplication is one of four main operations in elementary arithmetic, and most people learn basic multiplication algorithms in elementary school. The inverse of multiplication is division.

1 Multiplication of Integers
2 Properties
3 Division
- 3.1 Notation

Multiplication of Integers

Multiplication is written using the multiplication sign "×" between the terms; that is, in infix notation. The result is expressed with an equal sign. For example,

$2 \times 3 = 6$

$3 \times 4 = 12$

$2 \times 3 \times 5 = 30$

$2 \times 2 \times 2 \times 2 \times 2 = 32$

There are several other common notations for multiplication:

Multiplication is sometimes denoted by either a middle dot:

$5 \cdot 2 = 10$

In general, when multiplying integers where negative umbers are involved, the process is the same, with one exception. Count the number of negative signs in the problem. If there is an even number of negative signs, then your answer will be positive. If there is an odd number of negative signs, then the answer will be negative. This is sometimes called the Even-and-Odd Rule. For example,

$(-1) \cdot (-2) \cdot (-3) = -6$

Since the number of negatives in the example above is three (3), and three is an odd number, the answer will be negative (in this case, -6). However, if the number of negatives is an even number, for example,

$(-1) \cdot (-2) \cdot (-3) \cdot (-4) = 24$

then the answer is positive, since there are four (4) negative signs in the problem.

This Even-and-Odd Rule works in all cases and at all times, regardless of whether we are dealing with whole numbers, integers, real numbers, or otherwise.

Properties

For integers, fractions, real and complex numbers, multiplication has certain properties:

Commutative property

The order in which two numbers are multiplied does not matter.

x · y = y · x.

Associative property

Problems solely involving multiplication are invariant with respect to Order of Operations.

(x · y)·z = x·(y · z).

Distributive property

Holds with respect to addition over multiplication. This identity is of prime importance in simplifying algebraic expressions.

x·(y + z) = x·y + x·z.

Identity element

of multiplication is 1; anything multiplied by one is itself. This is known as the identity property

x · 1 = x.

Zero element

Anything multiplied by zero is zero. This is known as the zero property of multiplication.

x · 0 = 0

Inverse property

Every number x, except zero, has a multiplicative inverse, 1/x, such that x·(1/x) = 1.

Division

In Mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication.

Specifically, if c times b equals a, written:

$c \times b = a\,$

where b is not zero, then a divided by b equals c, written:

$\frac ab = c$

For instance,

$\frac 63 = 2$

since

$2 \times 3 = 6\,$ .

In the above expression, a is called the dividend, b the divisor and c the quotient.

Note: Division by zero (i.e. where the divisor is zero) is not defined.

Notation

Division is most often shown by placing the dividend over the divisor with a horizontal line, also called a vinculum, between them. For example, a divided by b is written

$\frac ab.$

This can be read out loud as "a divided by b" or "a over b". A way to express division all on one line is to write the dividend, then a slash, then the divisor, like this:

$a/b.\,$

Any of these forms can be used to display a fraction. A fraction is a division expression where both dividend and divisor are integers (although typically called the numerator and denominator), and there is no implication that the division needs to be evaluated further.

A less common way to show division is to use the obelus (or division sign) in this manner:

$a \div b.$

This form is infrequent except in elementary arithmetic. The obelus is also used alone to represent the division operation itself, as for instance as a label on a key of a calculator.

In general, division can be thought of as nothing more than multiplication of fractions. For instance, fractions such as ${a \over b}$ are typically defined as $a \cdot {1 \over b}$ .