3.1 Natural numbers, whole numbers and integers

3.4 Construction of rational and irrational numbers in a real line

Constructing Rational and Irrational Numbers on the Real Line involves visualizing and representing these numbers accurately. Here’s a detailed overview of how both types of numbers can be constructed and represented on the real line:

1. Construction of Rational Numbers

Rational Numbers can be represented on the real line by placing fractions at appropriate positions. The construction involves:

Steps

1. Identify the Fraction: A rational number pq\frac{p}{q}qp​ is where $p$ and $q$ are integers and q≠0q \neq 0q=0.
2. Divide the Line: To locate pq\frac{p}{q}qp​ on the real line:
   • Start with 0: Mark the point for 0 on the real line.
   • Mark 1 Unit: Determine the point for 1 unit on the real line.
   • Divide the Interval: Divide the interval between 0 and 1 into $q$ equal parts.
   • Locate the Fraction: Move $p$ parts from 0 to locate pq\frac{p}{q}qp​.

Example

To locate 23\frac{2}{3}32​ on the real line:

• Divide the interval between 0 and 1 into 3 equal parts.
• Mark the point that is 2 parts from 0. This point represents 23\frac{2}{3}32​.

2. Construction of Irrational Numbers

Irrational Numbers cannot be represented exactly as a simple fraction and their decimal expansions are non-terminating and non-repeating. Constructing them on the real line typically involves approximation or geometric methods.

Steps

1. Approximation: Use decimal approximations to represent irrational numbers on the real line.
   • For example, $\pi$ is approximately 3.14159. Locate this approximation on the real line.
2. Geometric Construction:
   • Square Root of a Number: To construct 2\sqrt{2}2​:
     • Start with a line segment of length 1.
     • Construct a right triangle with both legs of length 1.
     • The hypotenuse of this triangle represents 2\sqrt{2}2​.
   • Example:
     • Draw a line segment of length 1.
     • From each endpoint, draw a perpendicular line segment of length 1.
     • Connect the endpoints of the perpendicular lines to form a right triangle.
     • The length of the hypotenuse will be 2\sqrt{2}2​.

3. Representation on the Real Line

• Rational Numbers: They can be exactly represented as points on the real line where their corresponding fractions are located.
• Irrational Numbers: They are often represented as points on the real line based on decimal approximations or through geometric constructions. Because they cannot be precisely located using a finite decimal or fraction, approximations are used for practical purposes.

3.5 Real numbers and its properties :

Real Numbers encompass all the numbers that can be represented on the number line, including rational and irrational numbers. They have several important properties that facilitate arithmetic operations and mathematical reasoning. Here’s an overview of key properties of real numbers:

1. Addition Property

The Addition Property of real numbers describes the behavior of addition with respect to real numbers.

Properties

• Commutative Property: For any real numbers $a$ and $b$,

  $$a+b=b+a$$The order in which numbers are added does not affect the sum.

• Associative Property: For any real numbers $a$, $b$, and $c$,

  $$(a+b)+c=a+(b+c)$$The grouping of numbers being added does not affect the sum.

• Identity Property: There exists an additive identity $0$ such that for any real number $a$,

  $$a+0=a$$Adding zero to a number does not change its value.

2. Multiplication Property

The Multiplication Property of real numbers describes the behavior of multiplication with respect to real numbers.

Properties

• Commutative Property: For any real numbers $a$ and $b$,

  $$a\times b=b\times a$$The order in which numbers are multiplied does not affect the product.

• Associative Property: For any real numbers $a$, $b$, and $c$,

  $$(a\times b)\times c=a\times (b\times c)$$The grouping of numbers being multiplied does not affect the product.

• Distributive Property: For any real numbers $a$, $b$, and $c$,

  $$a\times (b+c)=(a\times b)+(a\times c)$$Multiplication distributes over addition.

• Identity Property: There exists a multiplicative identity $1$ such that for any real number $a$,

  $$a\times 1=a$$Multiplying a number by one does not change its value.

3. Distributive Property

The Distributive Property links addition and multiplication and is crucial for simplifying expressions and solving equations.

Property

• Distributive Law: For any real numbers $a$, $b$, and $c$, $$a\times (b+c)=(a\times b)+(a\times c)$$ This property shows that multiplication distributes over addition, allowing you to multiply each term in a sum separately and then add the results.

4. Density Property

The Density Property refers to the idea that between any two real numbers, there is always another real number. This property is essential in understanding the continuity of the real number line.

Property

• Density of Real Numbers: For any two real numbers $a$ and $b$ where $a<b$, there exists another real number $c$ such that $a<c<b$.
  • Example: Between any two real numbers like 1 and 2, you can find 1.5, or even more precisely, 1.01, 1.001, etc. This demonstrates that the real numbers are dense.

3.5 Absolute value of real numbers

The absolute value of a real number is a fundamental concept in mathematics that measures the distance of the number from zero on the real number line, regardless of direction.

Definition

For a real number $x$, the absolute value is denoted as $|x|$ and is defined as follows:

• If $x$ is a non-negative number (i.e., $x\ge 0$), then $|x|=x$.
• If $x$ is a negative number (i.e., $x<0$), then $|x|=-x$, which is the positive value of $x$.

In other words, the absolute value of a number is always non-negative.

Mathematical Definition

∣x∣={xif x≥0−xif x<0|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}∣x∣={x−x​if x≥0if x<0​

Examples

1. Positive Number:

   $$|5|=5$$Here, 5 is already non-negative, so its absolute value is itself.

2. Negative Number:

   $$|-3|=3$$Here, -3 is negative, so its absolute value is the positive counterpart of -3.

3. Zero:

   $$|0|=0$$Zero is neither positive nor negative, so its absolute value is zero.

Properties of Absolute Value

1. Non-Negativity:

   $$|x|\ge 0\text{ for all real numbers }x.$$

2. Identity Property:

   $$|x|=0\text{ if and only if }x=0.$$

3. Symmetry:

   $$|x|=|-x|.$$The absolute value of $x$ is the same as the absolute value of its negative.

4. Triangle Inequality:

   $$|x+y|\le |x|+|y|.$$The absolute value of the sum of two numbers is less than or equal to the sum of their absolute values.

5. Multiplication Property:

   $$|x\cdot y|=|x|\cdot |y|.$$The absolute value of the product of two numbers is the product of their absolute values.

6. Division Property:

   ∣xy∣=∣x∣∣y∣\left| \frac{x}{y} \right| = \frac{|x|}{|y|}​yx​​=∣y∣∣x∣​Provided y≠0y \neq 0y=0. The absolute value of the quotient is the quotient of the absolute values.

Graphical Interpretation

On the real number line, the absolute value of a number $x$ represents its distance from the origin (0). For example, the absolute value of both 3 and -3 is 3 because both are 3 units away from 0.

Applications

• Distance Measurement: Absolute value measures the distance between numbers, which is useful in various applications including geometry and physics.
• Error Measurement: In numerical analysis, the absolute value is used to measure the magnitude of errors.
• Equations and Inequalities: Absolute value is used in solving equations and inequalities involving distances and deviations.