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Square Root. Real Numbers.

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Square root

A number whose square is equal to \(a\) is called the square root of \(a\).

Real numbers

Real numbers are a type of number used in mathematics that include all rational and irrational numbers. They are represented by the symbol "\(R\)" and are used to represent quantities that can be measured, counted or calculated.
Rational numbers are numbers that can be expressed as the ratio of two integers, such as \(\frac{2}{3}\) or \(-\frac{4}{7}\). Irrational numbers, on the other hand, are numbers that cannot be expressed as the ratio of two integers and have an infinite number of non-repeating decimals, such as pi (\(\pi\)) or the square root of (\(\sqrt{2}\)).
Together, rational and irrational numbers make up the set of real numbers. Real numbers are typically represented on a number line, which is a horizontal line with a zero point in the center, and negative numbers on the left and positive numbers on the right.
Real numbers are used in various mathematical concepts, such as algebra, calculus, and geometry, and are essential in many scientific and engineering applications. They are also used in everyday life, such as in measurements of distance, time, temperature, and weight.

Properties of real numbers.

Rational Numbers

In mathematics, a rational number is a number that can be expressed as the ratio of two integers, where the denominator is not equal to zero. In other words, a rational number is a fraction where the numerator and denominator are both integers.
For example: \(\frac{2}{3}\), \(\frac{5}{8}\) and \(-\frac{7}{11}\) are all rational numbers. However, numbers such as \(\sqrt{2}\) or \(\pi\) (pi) are not rational, because they cannot be expressed as a ratio of two integers.
Rational numbers have some important properties, including closure under addition, subtraction, multiplication, and division. This means that if you add, subtract, multiply, or divide two rational numbers, the result will also be a rational number.
Rational numbers also have a decimal expansion, which can be either finite or repeating. For example, \(\frac{2}{5}\) can be written as a decimal \(0.4\), and \(\frac{1}{3}\) can be written as a repeating decimal \(0.333…\) The decimal expansion of a rational number can be obtained by dividing the numerator by the denominator.
In addition, the set of rational numbers is dense in the real number line, which means that between any two distinct rational numbers, there exists another rational number. This property is useful for approximating real numbers with rational numbers.

Irrational Numbers.

In mathematics, an irrational number is a real number that cannot be expressed as a ratio of two integers, or as a repeating or terminating decimal. Irrational numbers are infinite non-repeating decimal numbers that cannot be expressed exactly as a fraction.
For example, pi \( ( \pi ) \), the square root of 2 \( ( \sqrt{2} ) \), and \(e\) (the base of the natural logarithm) are all examples of irrational numbers. Unlike rational numbers, which have a finite or repeating decimal representation, irrational numbers have an infinite, non-repeating decimal expansion.
The decimal representation of an irrational number can be calculated to any desired precision, but it will never terminate or repeat. For example, the value of pi can be approximated to any desired number of decimal places, but it will never be expressed exactly as a ratio of two integers.
Irrational numbers have some important properties. They are closed under addition, subtraction, and multiplication, which means that if you add, subtract, or multiply two irrational numbers, the result will also be irrational. However, when an irrational number is added to a rational number, the result is always an irrational number.
The set of irrational numbers, together with the set of rational numbers, forms the set of real numbers. The real numbers are used extensively in calculus, analysis, and other branches of mathematics.


In mathematics, integers are a set of whole numbers that includes zero, the positive whole numbers \( (1,2,3,...) \), and the negative whole numbers \( (-1,-2,-3,...) \). Integers are denoted by the symbol "\(Z\)" and are represented on the number line as equally spaced points in both the positive and negative directions.

Natural numbers.

In mathematics, natural numbers are the set of positive integers \( (1,2,3,...) \) that are used to count or label objects. Natural numbers are denoted by the symbol "\(N\)" and are a subset of the set of integers.

Quadratic function

The function \(y=x^2\) is a second-degree polynomial function that maps every real number \(x\) to its square, or the product of \(x\) with itself. In other words, the value of \(y\) is equal to the square of the input \(x\).
The graph of this function is a parabola that opens upwards, and it is symmetric with respect to the \(y\)-axis. The vertex of the parabola is at the origin \( (0,0) \), and as \(x\) increases or decreases from zero, the value of \(y\) increases as well.

Here are some properties of the function \(y=x^2\):

The function \(y = \sqrt{x} \)

The function \(y = \sqrt{x}\) is the square root function, which maps the non-negative real numbers onto their square roots. In other words, for any non-negative value of \(x\), the square root of \(x\) (which is always positive) is the value of \(y\).

Here are some key features of the square root function:


Exponents are a shorthand notation for writing repeated multiplication of a number or expression by itself. An exponent is a small number or symbol that is written above and to the right of a base number or expression. The exponent indicates how many times the base is multiplied by itself.
The basic format for writing an exponent is: \( a^n \)
Base number or expression raised to the power of an exponent.
For example, 2 raised to the power of 3 (written as \( 2^3 \) means 2 multiplied by itself three times:
\(2^3=2\cdot 2\cdot 2=8.\)

Here are some key concepts and rules associated with exponents: \(a \neq 0, b \neq 0 \).

Exponential functions:

An exponential function is a function of the form \( f(x) = a^x \), where \(a\) is a positive constant called the base of the function. The value of \(a\) determines the shape of the graph of the function. Exponential functions grow or decay at a constant rate, which is determined by the value of \(a\).

Scientific notation:

Scientific notation is a way of writing very large or very small numbers using exponents. In scientific notation, a number is written as a decimal number between \(1\) and \(10\), multiplied by a power of \(10\). For example, the number \( 3,000,000 \) can be written as \( 3 \cdot 10^6 \), and the number \( 0.00005 \) can be written as \( 5 \cdot 10^{-5} \).
Exponents are a fundamental concept in mathematics, and they have many practical applications in fields such as science, engineering, finance, and computer science.