Arithmetic is a branch of mathematics that is responsible for studying the elementary number structures, as well as the properties of operations and numbers.For you it is easier to find the arithmetic in your life when:

- You go to the store to buy something, and you see in the need to calculate by means of a subtraction, the change that the shopkeeper will give.
- When you are about to approach the public service and how quickly the amount of money needed to pay the value of the ticket.
- also when you do the account or inventory of your things.

Formal arithmetic education usually begins in kindergarten and is completed by the end of fourth grade, and it is normally integrated with other subjects such as science, history and literature by teaching students to tell time, learn dates and solve story problems with numbers. Kindergartners usually learn to count from one to twenty, forwards and backwards and to pick up the count from any number.

They also learn comparative concepts such as larger than smaller than and to add and subtract numbers up to five. Most kindergartners also learn the use of balance scales, measuring instruments and pattern blocks,plastic cubes, pumpkin seeds, acorns and other materials for weighing, counting, comparing, sorting and understanding that written and spoken numbers are conceptually the same as the equivalent number of objects.

**First grade arithmetic** progresses to include the following: counting forward and backward from 1 to 100; skip counting by twos, fives and tens; instant identification of numbers before and after any number from 1 to 100, the concept of place values; simple addition and sub- traction of all single- and double-digit numbers; recognition of the inverse relationship between addition and subtraction; the use of the equality sign (=); the concept of fractions as parts of a whole; and the use of numbers in daily life to measure money, time, capacity, weight and temperature. First graders also extend their learning beyond arithmetic into other branches of mathematics by learning to recognize two- dimensional geometric figures.

**In second grade**, students learn to master numeration to 1,000 — adding and subtracting three single- and two-digit numbers and “rounding off” to the nearest ten or hundred. The standard curriculum also includes counting by odd and even numbers and to 1,000 by twos, threes, fives and tens. Measurement skills include mastery of money, time and the calendar.

**Third graders** progress to mastery of the multiplication tables through 12. They also learn to add and subtract four- and five-digit numbers, to multiply or divide two- and three- digit numbers by single-digit multiplicands and divisors and to recognize the inverse relation- ship between multiplication and division. Calculation skills also include mastery of fractions, decimals and the ability to convert from one to the other.

## Applications of Arithmetic In Basic Mathematics Which Every Mathematicians Must Know

**What Are The Natural Numbers?**

Natural number, which serves to designate the number of elements that have a certain set, and is called cardinal of said set.The natural numbers are infinite. The set of all of them is designated by N:

N = {0, 1, 2, 3, 4, …, 10, 11, 12, …}

**Properties of the addition of Natural Numbers**

The addition of natural numbers fulfills the properties associative, commutative and neutral element.

1.- Associative:

If a, b, c are any natural numbers, it follows that:

(a + b) + c = a + (b + c)

For example:

(7 + 4) + 5 = 11 + 5 = 16

7 + (4 + 5) = 7 + 9 = 16

The results coincide, that is,

(7 + 4) + 5 = 7 + (4 + 5)

2.-Commutative

If a, b are any natural numbers, it follows that:

a + b = b + a

In particular, for numbers 7 and 4, it is verified that:

7 + 4 = 4 + 7

Thanks to the associative and commutative properties of the addition, long sums of natural numbers can be made without using parentheses and without taking into account the order.

3.- Neutral element

The 0 is the neutral element of the sum of integers because, whatever the natural number a, it is fulfilled that:

a + 0 = a

**Properties of the Multiplication of Natural Numbers**

The multiplication of natural numbers fulfills the associative, commutative, neutral and distributive properties of the product with respect to addition.

1.-Associative

If a, b, c are any natural numbers, it follows that:

(a · b) · c = a · (b · c)

For example:

(3 · 5) · 2 = 15 · 2 = 30

3 · (5 · 2) = 3 · 10 = 30

The results coincide, that is,

(3 · 5) · 2 = 3 · (5 · 2)

2.- Commutative

If a, b are any natural numbers, it follows that:

a · b = b · a

For example:

5 · 8 = 8 · 5 = 40

3.-Neutral element

The 1 is the neutral element of the multiplication because, whatever the natural number a, it is fulfilled that:

a · 1 = a

4.- Distributive of the product with respect to the sum

If a, b, c are any natural numbers, it follows that:

a · (b + c) = a · b + a · c

For example:

5 · (3 + 8) = 5 · 11 = 55

5 · 3 + 5 · 8 = 15 + 40 = 55

The results coincide, that is,

5 · (3 + 8) = 5 · 3 + 5 · 8

**Properties of the Subtraction of Natural Numbers**

Like the sum the subtraction is an operation that is derived from the counting operation.

If we have 6 sheep and the wolves eat 2 sheep, how many sheep do we have? One way of doing this would be to recount all the sheep, but someone who had told the same case several times would remember the result and would not need to count the sheep again. I would know that 6 – 2 = 4.

The terms of the subtraction are called minuendo (the sheep that we have) and subtracted (the sheep that were eaten by the wolves).Subtraction properties:

Subtraction does not have the commutative property (it is not the same a – b as b – a)

**Properties of the Division of Natural Numbers**

The division is the operation that we have to do to distribute a number of things among a number of people.

The terms of the division are called dividends (the number of things), divisor (the number of people), quotient (the number that corresponds to each person) and rest (what is left over).

If the rest is zero, the division is called exact and otherwise inaccurate.

#### Properties of the division

The division does not have the commutative property. It is not the same a / b as b / a.

### What are the Whole Numbers?

Integer, any element of the set formed by the natural numbers and their opposites. The set of integers is designated by Z:

Z = {…, -11, -10, …, -2, -1, -0, 1, 2, …, 10, 11, …}

Negative numbers allow you to count new types of amounts (such as debit balances) and order above or below a certain reference item (temperatures above or below 0 degrees, the floors of a building above or below the entrance to it …).

It is called the absolute value of an integer a, a natural number that is designated | a | and that is equal to one’s own if it is positive or zero, and -a if it is negative. That is to say:

• if a> 0, | a | = a; for example, | 5 | = 5; • if a <0, | a | = -a; for example, | -5 | = – (- 5) = 5.

The absolute value of a number is, therefore, always positive.

The addition, subtraction and multiplication operations of whole numbers are internal operations because their result is also a whole number. However, two integers can only be divided if the dividend is a multiple of the divisor.

To add two whole numbers, proceed as follows:

• If they have the same sign, their absolute values are added, and the result is given the sign that the addendums had: • 7 + 11 = 18 • -7 – 11 = -18 • If they have different signs, that is, if one addition is positive and the other negative, its absolute values are subtracted and the sign of the greater is given: • 7 + (-5) = 7 – 5 = 2 • -7 + 5 = – (7 – 5) = – 2 • 14 + (-14) = 0

The sum of integers has the following properties:

Associative: (a + b) + c = a + (b + c) Commutative: a + b = b + a Neutral element: zero is the neutral element of the sum, a + 0 = a Opposite element: any whole number a, has an opposite -a, a + (-a) = 0

**Multiplication of Whole Numbers**

To multiply two whole numbers multiply their absolute values and the result is left with a positive sign if both factors are of the same sign or the minus sign is given if the factors are of different signs. This procedure to obtain the sign of a product from the sign of the factors is called the rule of the signs and is synthesized as follows:

+ · + = + + · – = – – · + = – – · – = +

The multiplication of integers has the following properties:

Associative: (a · b) · c = a · (b · c) Commutative: a · b = b · a Neutral element: 1 is the neutral element of multiplication, a · 1 = a Distributive of multiplication with respect to the sum: a · (b + c) = a · b + a · c

To subtract two whole numbers, the minuend is added to the opposite of the subtrahend:

a – b = a + (-b)

For example:

5 – (-3) = 5 + 3 = 8 -2 – 5 = (-2) + (-5) = -7

**What are the Fractional Numbers?**

The Fractional Numbers, are the quotient indicated a / b of two integers that are called numerator, a, and denominator, b. It has to be b ≠ 0.

For example, in fraction 3/5 the denominator, 5, indicates that they are “fifth parts”, that is, it calls the type of part of the unit in question; The numerator, 3, indicates how many of these parts should be taken: “three fifths”.

If the numerator is a multiple of the denominator, the fraction represents an integer:

14/2 = 7; -15 / 3 = -5; 352/11 = 32

**Equivalence**

Two fractions a / b and a ‘/ b’ are equivalent, and it is expressed

a / b = a ‘/ b’ if a · b ‘= b · a’.

A) Yes,

21/28 = 9/12 because 21 · 12 = 9 · 28 = 252.

**Simplification**

If the numerator and the denominator of a fraction are divisible by the same number, d, other than 1 or -1, dividing them by d gives another fraction equivalent to it. It is said that the fraction has been simplified or reduced: a / b = a.d ‘/ b.d’ = a ‘/ b’

For example: 120/90 = 12/9

The 12/9 fraction is the result of simplifying 120/90 by dividing its terms by 10

**Irreducible fractionation**

It is said that a fraction is irreducible if its numerator and denominator are prime numbers together.

The fraction 3/5 is irreducible. The fraction 12/9 is not irreducible because it can be simplified: 12 / = 4/3

**Reduction to common denominator**

To reduce two or more fractions to a common denominator is to obtain other fractions respectively equivalent to them and that all have the same denominator. If the fractions from which they are split are irreducible, the common denominator must be a common multiple of their denominators. If it is the least common multiple (mcm) of them, then it is said that it has been reduced to the lowest common denominator.

For example, to reduce common denominator fractions

2/3, 4/9 and 3/5

you can take 90 as the common denominator, with what you get: 2/3 = 60/90, 4/9 = 40/90, 3/5 = 54/90

That is, it is the result of reducing the three previous fractions to a common denominator: 90.

But if instead of 90 is taken as common denominator 45, which is the LCM of 3, 9 and 5, then you get

30/45, 20/45, 27/445

which is the result of reducing the three fractions to their lowest common denominator.

**Sum of Fractions**

To add two or more fractions are reduced to a common denominator, the numerators are added and their denominator is maintained. For example: 2/3 + 4/9 and + 3/5 = 30/45 + 20/45 + 27/45 = 30 + 20 + 27/45 = 77/45

**Fractions Product**

The product of two fractions is another fraction whose numerator is the product of its numerators and whose denominator is the product of its denominators: a / b * c / d = a * c / b * d

**Reverse of a Fraction**

The inverse of a fraction a / b is another fraction, b / a, which is obtained by permuting the numerator and the denominator. The product of a fraction by its inverse is equal to 1:

a / b * b / a = a * b / b * a = 1/1 = 1

**Fraction Quotient**

The quotient of two fractions is the product of the first by the inverse of the second: a / b: p / q, a / b * q / p, a * q / b * p

Decimal number, any fractional number expressed in the decimal numbering system. Thus, the numbers 7,84; 0.005; -2.8464646 …; 3,141592 … it is said that they are decimals.

The concept of positional values, which is fundamental to expressing integers by Arabic notation on any number basis, can be extended to include fractional numbers. If the numbering base is 10, the different numbers of a whole number mean its value multiplied by a positive power of 10:

3.586 = 3 · 10 ^ 3 + 5 · 10 ^ 2 + 8 · 10 ^ 1 + 6 · 100

To include the numbers with fractional parts you must include figures with their values multiplied by negative powers of 10. These figures are placed to the right of the units separated by a comma:

127,546 = 1 * 10 ^ 2 + 2 * 10 ^ 1 + 7 * 10 ^ 0 + 5 * 10 ^ -1 + 4 * 10 ^ -2 + 6 * 10 ^ -3 =

1 * 100 + 2 * 10 + 7 + 5 * 1/10 + 4 * 1/100 + 6 * 1/1000

The fractional units to the right of the comma are called tenths, hundredths, thousandths, ten thousandths, …, millionths.

If a decimal number has a finite number of decimal places, it is usually called exact decimal and corresponds to an irreducible fraction whose denominator decomposed into prime factors only has factors 2 and 5.

For example: 5,42 = 542/100

Because the denominator is a power of 10, it only has factors 2 and 5, and by reducing the fraction, only these factors remain.

There are decimals with an infinite number of figures that are repeated periodically. They are called periodic decimals and are obtained from irreducible fractions whose denominator has a factor other than 2 or 5.

For example: 3.4222 ……. = 3.42 = 154/45

Finally, there are decimal numbers with infinite numbers that are not repeated periodically. They do not correspond to any fraction and, therefore, are irrational numbers. It is the case of

pi = 3.141592 …

square root of 2 = 1,41424 …

Decimal numbers can be represented on the real line: if they have a finite number of digits, they can be placed theoretically accurate; If their figures are infinite, they can be placed as close as desired