**Historical context of algebra**

Algebra is a branch of mathematics that provides simple solutions to many complex mathematical problems, especially when quantities are represented by a sign with no arithmetic value.

The word algebra comes from the Arabic word al-Jabar. The Persian Muslim mathematician Muhammad ibn Musa al-Khwarizmi wrote in 820 CE. Chr. the Kitab al-mukhtasar fi hisab al-gabr wa-l-muqabala in Arabic, in which he explains techniques for solving complex mathematical problems.

This book was then published in Europe under the title Algebra, so that Al-Jabar became Algebra.

A real song:

A real number is a value that can be represented along a numerical line as

Classification of real numbers :

The current figures are as follows,

- Rational numbers
- Fractions
- Integrators
- Whole numbers
- Negative integrals

- Irrational figures

We will now discuss all of the above real number types….. individually.

**1 – Rational numbers**

Rational numbers can be written as the ratio of two integers in the form p/q, where p and q are integers and q is not zero. That’s what question Q indicates. Some examples of rational numbers,

There are two ways to represent the rational numbers

decimally

- The end of the decimal numbers

is a fixed number of digits in the decimal part

. For example, for example..,

- Non-consecutive or repeating decimal numbers – there is no fixed number of digits in decimal

or some digits are repeated ad infinitum. For example, for example..,

**1.1 – Fractions**

A fraction can be spelled as,

For example, for example..,

Types of fractures ;

- Properties of fractions – In regular fractions the numerator is always smaller than the denominator, for example

- Irregular fractions – In irregular fractions, the numerator is always equal to or greater than the denominator, for example. B. .

- Mixed fractions – Mixed fractions have, for example, an actual fraction and an integer. B. .

**1.2 – Integrals**

It can be an integer or a negative integer

without decimals. Indicated by Z, i.e. .

**1.2.1 – Integer numbers**

Integer numbers are integers with a negative integer

, beginning with zero and continuing on a numeric line. The integers

are denoted by W, i.e.

Note here

that positive integers are like integers, but they start with 1 to

and then without 0, i.e.

**1.2.2 – Natural numbers**

Natural numbers are integers beginning with

from 1 to 1, i.e. 1, 2, 3, 4, 5,…………….. Natural numbers are denoted by the number

, i.e.

Zero:

Zero is considered an integer

, but has no positive or negative value.

Negative integrals :

Negative integers start at minus 1 and continue, i.e.

**2 – Irrational numbers:**

Irrational numbers are real numbers

, which cannot be written as a ratio of two integers. Marked Q. Some examples of irrational numbers

,

**Mathematical properties of real numbers**

The mathematical properties of real numbers

in terms of addition, subtraction, multiplication and division are :

**Additional features:**

Closure of properties :

If a, b are two real numbers and a + b = c, then c –

is also an integer.

5 + 3 = 8. Where 8 is also an integer.

Community property :

If a, b are two real numbers,

a + b =

b + a.

For example, for example..,

a = 5, b = 3

⇒ 5 + 3 = 8 = 3 + 5

Associated Property:

Let a, b, c be 3 integers,

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

For example, for example..,

a = 3, b = 5, c = 4.

3 + (5 + 4) = (3 + 5) + 4 = (3 + 4) + 5 = 12

Additive identity :

Thus, in the additive identification, there is a separate real number

, namely 0,

a + 0 = 0 + a

For example, for example..,

3 + 0 = 3 = 0 + 3

Additive reversal :

The additive reciprocal of a is denoted by – a, then

a + (-a) = 0 = (-a) + a

This means that the inverse additive is 1 – 1. For example, for example..,

5 + (–5) = 0 = (–5) + 5

**Subtraction:**

Closure of properties :

If a, b are two real numbers and a – b = c, then c –

is not always an integer. For example, for example..,

So, a = 5 and b = 3, then …..

5 – 3 = 2 (integer) then how,

3 – 5 = -2, which is not a whole number.

Community property :

If a, b are two real numbers,

For example, for example..,

a = 5, b = 3

Associated Property:

Let a, b, c be 3 integers,

For example, for example..,

a = 3, b = 5, c = 4.

### Propagation properties

Closure of properties :

If a, b are two integers and , then c is also an integer. For example, for example..,

So, a = 5 and b = 3, then …..

(integer

) and

, i.e.

is also an integer.

Community property :

If a, b are two real numbers,

For example, for example..,

a = 5, b = 3

Associated Property:

Suppose a, b, c are three integers,

For example, for example..,

a = 3, b = 5, c = 4.

The multiplier is equal to zero :

The multiplication property is equal to zero,

For example, for example..,

Multiplicative identification :

A multiplicative identification is given,

For example, for example..,

**Real Estate Sector:**

Closure of properties :

This property indicates that the result of dividing two numbers

is not always an integer. For example, for example..,

(integer

), where,

Community property :

If a, b are two real numbers,

For example, for example..,

a = 20, b = 10

Associated Property:

Suppose a, b, c are three integers,

For example, for example..,

a = 3, b = 5, c = 4.

You might also be interested: Types of equations

**Arithmetic operations**

Order of operations :

Basic arithmetic operations

:

- the addition of
- Subtract
- Multiplication
- Department

It is relevant to note here that if one arithmetic expression

contains more than one arithmetic operation, it is necessary for

to understand the order of the arithmetic operations.

To ensure the proper execution of orders, the following rules generally apply.

- Operations

within a parent must be performed first

. - Operations

of exponents and roots must be performed before propagation and division

. - Multiplication and division are performed earlier for addition and subtraction

. - Addition and subtraction should be performed from left to right to

.

One way to remember these sequences of arithmetic operations is

, an acronym formed from the red and bold letters of the rules above.

PEMDAS (parentheses, exponents, multiplication, division, addition, subtraction).

Another way to remember these sequences of arithmetic operations is to use this theorem

.

Excuse my dear Auntie Shahida.

Example 1 :

2 + 3 + (5 + 2)

Therefore, rule No. 1 should be applied here (operations within the Board of Investment should be implemented first),

2 + 3 + 7 = 12

Example 2 :

2 × 32

Rule No. 2 (operations on exponents

and roots should be performed earlier in multiplication and division) should apply here, so

,

2 x 9 = 18

Example 3 :

2 x 3 + 2

Therefore, rule number 3 must be applied here (multiplication and division of

are performed before addition and subtraction of

),

6 + 2 = 8

Example 4 :

2 + 3 – 2 + 5

Therefore, rule number 4 must apply here (addition and subtraction must be from left to right)

,

5 – 2 + 5

3 + 5 = 8

Example 5 :

2 3 + (2 × 32) – 3

Here you can apply the PEMDAS rule,

2 3 + (2 × 9) – 3

2 3 + 18 – 3

6 + 18 – 3

24 – 3 = 21

Positive and negative figures :

As we have already read, a system of real numbers contains a set of integers. An integer can be a whole number or a negative integer without decimals.

Here you need to understand the rules for adding, subtracting, multiplying and dividing positive and negative numbers. The following tables and examples facilitate the understanding of these processes.

The first value | Operation | The second value | Corresponds to | Terminal value |

+ | × | + | = | + |

+ | × | – | = | – |

– | × | – | = | + |

– | × | + | = | – |

+ | ÷ | + | = | + |

+ | ÷ | – | = | – |

– | ÷ | – | = | + |

– | ÷ | + | = | – |

Example 1 :

2 3 = 6

Example 2 :

2 (-3) = -6

Example 3 :

-2 (-3) = 6

Example 4 :

12 ÷ 3 = 4

Example 5 :

12 ÷ (-3) = -4

The first value | Operation | Second value
(less than or equal to |
equals to |
Terminal value |

+ | + | (Less) | = | + |

+ | + | – (Lake) |
= | – |

– | + | + (min) |
= | – |

– | + | + (Plus) |
= | + |

– | – | – (smaller) |
= | – |

– | – | – (Lake) |
= | + |

Example 1 :

4 + (-2) = 4 – 2 = 2

Example 2 :

2 + (-3) = 2 – 3 = -1

Example 3 :

2 + (-5) = 2 – 5 = -3

Example 4 :

12 + (-3) = 12 – 3 = 9

Example 5 :

12 – (-3) = 12 + 3 = 15

Example 6 :

2 – (-3) = 2 + 3 = 5

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