Square Root. Real Numbers.

Square root ☰

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

• Square root of a positive number \(a\) is written as \(\sqrt{x}\)
• Square root of a negative number is undefined in the real number system. It is denoted as \(\sqrt{-a}\)
• Multiplicative Property: The square root of the product of two numbers is equal to the product of their square roots.
  Mathematically \(\sqrt{ab}=\sqrt{a}\sqrt{b}\) .
• The square root of a quotient is equal to the quotient of the square roots of the numerator and denominator:
  \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\)
• The only number whose square is equal to zero is "0". That is, the square root of Zero is zero.
  \(\sqrt{0}=0\)
• The square root of a power is equal to the power of the square root: \(\sqrt[m]{a^n} = a^{\frac{n}{m}}\)

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.

• Closure property: The sum or product of any two real numbers is always a real number.
• Commutative property: The order of the terms does not affect the sum or product of two real numbers. That is, \(a+b=b+a\) and \(ab=ba\) for any real numbers \(a\) and \(b\)
• Associative property: The grouping of terms does not affect the sum or product of three or more real numbers.
  That is, \((a+b)+c=a+(b+c)\) and \((ab)c=a(bc)\) for any real numbers \(a\), \(b\) and \(c\).
• Distributive property: Multiplication distributes over addition. That is, \(a(b+c)=ab+ac\) and \((a+b)c=ac+bc\) for any real numbers \(a\), \(b\), and \(c\).
• Identity element: The sum of a real number and 0 is the same real number. That is, \(a+0=a\) for any real number \(a\). The product of a real number and 1 is the same real number. That is, \(a \cdot 1 = a \) for any real number \(a\)
• Inverse element: The sum of a real number and its additive inverse (opposite) is 0. That is, \(a+(-a)=0\) for any real number \(a\). The product of a nonzero real number and its multiplicative inverse (reciprocal) is 1. That is, \(\frac{a\cdot1}{a}=1\) for any nonzero real number \(a\).
• Transitive property: If \(a < b\) and \(b < c\), then \(a < c\) for any real numbers \(a\), \(b\) and \(c\).
• Trichotomy property: For any two distinct real numbers \(a\) and \(b\), exactly one of the following holds: \(a < b \), \(a=b\) or \(a> b \).
• Archimedean property: For any two positive real numbers \(a\) and \(b\), there exists \(a\) natural number \(n\) such that \( na > b \)

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.

Integers. ☰

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 \(y=x^2\):

• Domain: The domain of the function is all real numbers, since any real number can be squared.
• Range: The range of the function is all non-negative real numbers, since the square of any real number is non-negative.
• Zeros: The function has only one zero at \(x=0\), since the square of any non-zero number is positive.
• Symmetry: The function is symmetric with respect to the \(y\)-axis, which means that replacing \(x\) with \(-x\) in the equation \( y=x^2 \) does not change the value of \(y\).
• Concavity: The function is concave upwards, which means that the rate of change of the function increases as \(x\) moves away from \(0\) in either direction.
• Second derivative: The second derivative of the function is a constant equal to 2, which is positive, indicating that the function is concave upwards and has a minimum at the vertex.
• Vertex Form: \( y=(x-h)^2 + k \). This is an alternate form for \( y=x^2 \) that can be useful when graphing the function or finding key features. In this form, the coordinates of the vertex are \( (h,k) \). For the function \( y=x^2 \), the vertex is at \( (0,0) \), so the vertex form would be \( y=(x–0)^2 +0 \), which simplifies to \(y=x^2 \).
• Factored Form: \( y=(x-a)(x+a) \). This is a way to factor the function \( y=x^2 \) into two binomials. In this form, the \(x\)-intercepts of the parabola are located at \( (a,0) \) and \( (-a,0) \). For the function \( y=x^2 \), the factored form would be \( y=(x-0)(x+0) \), which simplifies to \( y=x^2 \) .

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:

• Domain: The domain of the square root function is the set of non-negative real numbers, or \( [0, +\infty) \). This is because the square root of a negative number is not a real number.
• Range: The range of the square root function is also the set of non-negative real numbers, or \( [0, +\infty) \). This is because the square root of a non-negative number is always a non-negative number.
• Graph: The graph of the square root function is a curve that starts at the point \( (0,0) \) and increases gradually as \(x\) increases. The curve approaches the \(x\)-axis but never touches it, as the square root of 0 is 0, but the function is not defined for negative values of \(x\).
• Increasing: The square root function is an increasing function, which means that as \(x\) increases, so does \(y\).
  Some examples of input-output pairs for the square root function are:
  
  • \(x=0, y=\sqrt{0}=0\)
  • \(x=1, y=\sqrt{1}=1\)
  • \(x=4, y=\sqrt{4}=2\)
  • \(x=9, y=\sqrt{9}=3\)

Exponents.

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: \(a \neq 0, b \neq 0 \).

• Product rule: When multiplying two powers with the same base, add their exponents: \(a^m \cdot a^n=a^{m+n}\)
• Quotient rule: When dividing two powers with the same base, subtract their exponents: \(\frac{a^m}{a^n}=a^{m-n}\)
• Power rule: TWhen raising a power to another power, multiply their exponents: \((a^m)^n=a^{m\cdot n}\)
• Negative exponent rule: \(a^{-n} = \frac{1}{a^n}\)
• Zero exponent rule: \(a^0=1\)
• Product of powers rule: To find the power of a product, raise each factor to the power and multiply:
  \( (ab)^n= a^n \cdot b^n \)
• Quotient of powers rule: To find the power of a quotient, raise the numerator and denominator to the power and divide:
  \( ( \frac{a}{b} )^n = \frac{a^n}{b^n} \)

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.