Understand concepts involved in topic real numbers

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## Definitions and Examples in Real Numbers

- Real Numbers: Real numbers are a set of numbers that includes all rational and irrational numbers. They can be represented on the number line and include integers, fractions, decimals, and more. Real numbers are the most comprehensive set of numbers in mathematics.
- Rational Numbers: Rational numbers are numbers that can be expressed as a fraction p/q, where p and q are integers, and q is not equal to zero. In other words, rational numbers are those that can be written as the ratio of two integers. Examples include 1/2, -3/4, and 5.
- Irrational Numbers: Irrational numbers are numbers that cannot be expressed as a fraction of two integers. They have non-repeating, non-terminating decimal expansions. Common examples include the square root of 2 (√2), π (pi), and e (Euler’s number).
- Integers: Integers are whole numbers, both positive and negative, as well as zero. They do not include fractions or decimal points. Integers include numbers like -3, -2, -1, 0, 1, 2, 3, and so on.
- Whole Numbers: Whole numbers are a subset of integers that include all the positive integers (1, 2, 3, …) and zero (0). They do not include negative numbers.
- Natural Numbers: Natural numbers, also known as counting numbers, are a subset of whole numbers that begin at 1 and continue indefinitely (1, 2, 3, …). Natural numbers are used for counting and ordering.
- Imaginary numbers and infinity are not real numbers

## Proving √2 is irrational number

## Place Value

- Each place in a number has a different value called its place value. A place value chart is a useful way to summarize this information. The place values are separated into groups of three, called periods. The periods are
*ones, thousands, millions, billions, trillions*, and so on. In a written number, commas separate the periods.

### Place Value Chart and writing place value

### Using Place Value to Name Whole Numbers

Step 1. Starting at the digit on the left, name the number in each period, followed by the period name. Do not include the period name for the ones.

Step 2. Use commas in the number to separate the periods.

The number is eight trillion, one hundred sixty-five billion, four hundred thirty-two million, ninety-eight thousand, seven hundred ten.

### Using place value to Write Whole Numbers

Writing numbers given in words as digits:

Step 1. Identify the words that indicate periods. (Remember the ones period is never named.)

Step 2. Draw three blanks to indicate the number of places needed in each period. Separate the periods by commas.

Step 3. Name the number in each period and place the digits in the correct place value position.

Example : fifty-three million, four hundred one thousand, seven hundred forty-two

Identify the words that indicate periods.

Except for the first period, all other periods must have three places. Draw three blanks to indicate the number of places needed in each period. Separate the periods by commas.

Then write the digits in each period.

Put the numbers together, including the commas. The number is 53,401,742

## How to round a whole number to a specific place value?

Step 1. Locate the given place value. All digits to the left of that place value do not change unless the digit immediately to the left is 9, in which case it may. (See Step 3.)

Step 2. Underline the digit to the right of the given place value.

Step 3. Determine if this digit is greater than or equal to 5.

- Yes—add 1 to the digit in the given place value. If that digit is 9, replace it with 0 and add 1 to the digit immediately to its left. If that digit is also a 9, repeat.
- No—do not change the digit in the given place value.

Step 4. Replace all digits to the right of the given place value with zeros.

## Properties of Real Numbers

The properties of real numbers are fundamental concepts in mathematics, especially in algebra. These properties are used to perform operations and solve equations involving real numbers. Here’s a brief overview of the key properties:

1. **Commutative Property**: This property applies to addition and multiplication. It states that the order of the numbers does not change the result. For example, a + b = b + a and ab = ba.

2. **Associative Property**: This property also applies to addition and multiplication. It indicates that the way numbers are grouped does not affect the sum or product. For example, (a + b) + c = a + (b + c) and (ab)c = a(bc)

3. **Distributive Property**: This property links addition and multiplication. It states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. For example, a(b + c) = ab + ac

4. **Identity Property**: There are two types of identity properties:

– **Additive Identity**: The sum of any number and zero is the number itself. For example, a + 0 = a

– **Multiplicative Identity**: The product of any number and one is the number itself. For example, a × 1 = a

5. **Inverse Property**: This property also comes in two forms:

– **Additive Inverse**: The sum of a number and its opposite (negative) is zero. For example, a + (-a) = 0

– **Multiplicative Inverse**: The product of a number and its reciprocal is one. For example, a times 1/a = 1 (assuming a is not zero)

6. **Closure Property**: This property states that the sum or product of any two real numbers is a real number. For example, if a and b are real, then a + b and ab are also real.

7. **Zero Product Property**: If the product of two real numbers is zero, then at least one of the factors must be zero. For example, if ab = 0, then either a = 0 or b = 0 or both.

8. **Transitive Property**: If two numbers are equal to the same number, then they are equal to each other. For example, if a = b and b = c, then a = c

9. **Symmetric Property**: If one number is equal to another, then the second is equal to the first. For example, if a = b, then b = a.

10. **Reflexive Property**: Each number is equal to itself. For example, a = a

These properties are the building blocks for solving equations and understanding algebraic structures in mathematics. They apply not only to simple numbers but also to more complex expressions involving variables.

## Non terminating but repeating decimals

- Non-terminating but repeating decimals are decimal representations of rational numbers that, when divided out, result in a repeating pattern of digits after the decimal point. These numbers are often denoted with a bar over the repeating part. The repeating part of the decimal continues indefinitely, even though it’s a finite sequence of digits that repeats over and over. These are also known as “recurring decimals.”

- For example, the decimal representation of the fraction 1/3 is a non-terminating but repeating decimal:1/3 = 0.3333… (with the 3’s repeating indefinitely) Another common example is 1/7: 1/7 = 0.142857142857… (with the sequence “142857” repeating indefinitely)

- In general, any fraction p/q, where p and q are integers and q is not divisible by any prime other than 2 or 5 (in the denominator) will result in a non-terminating but repeating decimal when expressed in decimal form.

- These non-terminating but repeating decimals are a subset of rational numbers, as they can be expressed as fractions (ratios of integers), but their decimal representations exhibit a repeating pattern of digits after the decimal point.

## Prime, Composite, Even, Odd Numbers

- Prime Numbers : Prime numbers are natural numbers greater than 1 that are only divisible by 1 and themselves. Examples include 2, 3, 5, 7, and 11.

- Composite Numbers: Composite numbers are natural numbers greater than 1 that have more than two distinct positive divisors. In other words, they are not prime.
- Even Numbers: Even numbers are integers that are divisible by 2, meaning they have no remainder when divided by 2 (e.g., -4, -2, 0, 2, 4, …).
- Odd Numbers: Odd numbers are integers that are not divisible by 2, meaning they have a remainder of 1 when divided by 2 (e.g., -3, -1, 1, 3, 5, …).

## Complex and Imaginary Numbers

- Complex numbers are numbers of the form a + bi, where “a” and “b” are real numbers, and “i” is the imaginary unit, defined as the square root of -1.
- Imaginary numbers are numbers in the form of ai, where “a” is a real number, and “i” is the imaginary unit. They are used in complex numbers.

## Square Root of positive real numbers geometrically

For example, find the square root of 3.5 geometrically

- x is 3.5
- Draw a line AB of length x=3.5. From the point B, draw a line of 1 unit, where the length of BC=1
- Find the mid-point of AC and mark it D. Draw a semicircle with center D and radius AD. Draw a line perpendicular to AC passing through B and intersecting the semicircle at E.
- Then BE is square root of 3.5

## Finding root 2 plus root 5

## Applications of Real Numbers

Real numbers are versatile and play a crucial role in various applications. As you delve into the practical aspects of working with real numbers, you’ll find that they provide a solid foundation for understanding and solving problems in different contexts:

**Arithmetic Operations and PEMDAS:**- When tackling real-world problems, mastering the order of operations becomes essential. From calculating expenses to solving complex equations, understanding PEMDAS ensures accuracy in your computations.

**Representation with Fractions and Decimals:**- Real numbers extend beyond whole numbers, offering precision in representing quantities. Explore how fractions and decimals contribute to a nuanced understanding of numerical values.

**Percentage Calculations and Percents:**- Real numbers seamlessly integrate with percentages in practical scenarios. Whether you’re calculating discounts or interpreting growth rates, a solid grasp of percents enhances your ability to work with real-world data.

**Comparisons and Ratios:**- Ratios, expressions of relationships between quantities, are fundamental in real-world applications. Real numbers provide the foundation for making meaningful comparisons and understanding ratios in diverse contexts.

**Proportional Relationships and Proportions:**- Real numbers are instrumental in expressing proportional relationships. Whether you’re scaling recipes, understanding financial ratios, or solving engineering problems, proportions help maintain balance and consistency

**Prime Factorization:**- Explore the concept of prime factorization and its significance in breaking down numbers into their prime components. This technique is valuable in various mathematical and problem-solving scenarios.

**HCF and LCM:**- Real numbers play a crucial role in determining the HCF (Highest Common Factor) and LCM (Least Common Multiple) of sets of numbers. These concepts are essential in tasks ranging from simplifying fractions to time management calculations.