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MIPA : Mathematics

Name : Nur Hamid

Class Number : 6

ARITHMETIC

Arithmetic is one of important thing in Mathematics. It is widely discussed in

Mathematics. Its development is begun by the appearance of operators of arithmetic. The

famous operators in Arithmetic are quantifying, reduction, multiplication, and division.

The quantifying means increasing of one or more things. The result of quantifying must

be greater than before. For example, Andy has 3 books. Then, he buys 5 books again. Now,

Andy has 8 books. We can declare this situation in Arithmetic of Mathematic. We can say

that

3 + 5 = 8 (3 plus 5 equals 8),

where 3 is the first book that Andy has, 5 is the number of books that he buys, and 8 is the

book that Andy has now.

The reduction means decreasing of one or more things. If the quantifying makes amount

of thing increasing, so it makes the amount of thing decreasing. For example, Hamid has 10

pens. Because he is very kindhearted, he gives his friend 7 pens. Now, he has 3 pens. This

condition can be declared in Mathematics statement:

107 = 3 (10 minus 7 equals 3)

where 10 is pens which Hamid has, 7 is pens which Hamid gives, and 3 is pens which Hamid

has now.

Now, how about quantifying and reduction which are done repeatedly?

In arithmetic, there is an operator which declares the quantifying which is done

repeatedly. It is multiplication operator. Its symbol is x, but usually it is written .. For

example, we have the quantifying of 4 that is done three times. We can say in Mathematic

statement

4 + 4 + 4 = 12.

We can also declare this statement by multiplication. It is

4 x 3 = 12.

The statement above shows that 4 has quantified three times.

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Now only multiplication for repeatedly quantifying, arithmetic also has operator for

repeatedly reduction. It is division. It is symbolized by / and read divided. For example,

we have the division of 18 by 6 until the result is 0. We can declare in Mathematics statement

18666 =0.

The statement above shows that we need reduction three times until the result is 0. For this

situation, we can declare by division operator

18/6 =3.

Arithmetic also discusses lines of arithmetic and sum of arithmetic lines. It is very

important for our life.

Line of arithmetic is the line which contains the number which has increasing constantly.

One example of arithmetic line is

1, 4, 7, 10, (1)

The example above shows the line which contain number which has increasing constantly.

The first number of that line is 1, and increases constantly. The increasing of this number is 3.

We can call this increasing by difference.

In example (1), we can use some symbol to declare it. We can say u(1)=1, u(2)=4, and

so on. u(1)=1 has meant that the 1st

unit is 1, u(2) 2nd

4, and so on. For this example, we can

make formula for this example, this is u(n)=1+(n-1).3 where n is sequence of the number. We

are also able to make general form of arithmetic lines, it is

() ( ) Where : a = 1

stnumber

n = sequence of number

b = the difference of number

u(n) = unit to n.

Now, we must ask around about how the result of arithmetic line is. For this question,arithmetic also has side that is called by sum of arithmetic.

For any line arithmetic line, we can write the general form of sum of arithmetic, it is

() () () ()To count this result, we can use the general formula:

() ()

( ( )) ()

We use the example (1). We can count the sum of this line by the formula (2). If wecount the sum until n=10, we have line

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1, 4, 7, 10, 13, 16, 19, 22, 25, 28.

We may can quantify these number easily, but we will spend many times to solve it. For this,

we use

() ( ( ))

The proof of formula (2) is:

Let we have u(n)=a+(n-1)b for any line of arithmetic. We can write:

() ( )() ( ) ( )

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() ( ) ( ) ( )() ( ( ))() ( ( ))

The last form can be designed into

() ( ())

Where n is sequence of last number, a firs number, and u(n) the last unit of arithmetic line.