The set of natural numbers and Number bases

# The set of natural numbers and Number bases ## Number bases :Addition and subtraction in other bases ### Introduction Welcome to this lesson on performing addition and subtraction in number bases other than base 10! You've already learned about natural numbers and how to convert numbers between different bases. Now, we will take the next step by learning how to do basic arithmetic operations like addition and subtraction with numbers expressed in various bases. ### Key Concepts #### Addition in Other Bases When adding numbers in a specific base (e.g., base 2, base 5, base 8), the process is similar to base 10 addition, but with a crucial difference: any time a sum in a column is equal to or greater than the base, you must divide that sum by the base. The remainder is written down, and the quotient is carried over to the next column. * **Rule**: If the sum of digits in a column is equal to or greater than the base, divide the sum by the base. Write down the remainder and carry over the quotient to the next column. * **Process**: 1. Align the numbers vertically by their place values. 2. Start adding from the rightmost column (the lowest place value). 3. For each column: * Add the digits in that column (and any carried-over digit from the previous column). * If the sum is less than the base, write it down directly. * If the sum is equal to or greater than the base, divide the sum by the base. Write down the remainder below the column, and carry the quotient to the next column on the left. 4. Repeat this process for all columns, moving from right to left. #### Subtraction in Other Bases Subtracting numbers in other bases also follows a similar pattern to base 10 subtraction. The main difference occurs when you need to 'borrow' from a digit in a higher place value. When you borrow, the value you add to the current column is equal to the base itself, not 10. * **Rule**: When subtracting, if a digit in the top number is smaller than the digit below it, 'borrow' from the digit to its left. The digit you borrow from is reduced by 1, and the borrowed value added to the current digit is equal to the base. * **Process**: 1. Align the numbers vertically by their place values. 2. Start subtracting from the rightmost column. 3. For each column: * If the top digit is greater than or equal to the bottom digit, subtract normally and write down the result. * If the top digit is smaller than the bottom digit, you need to borrow: * Reduce the digit in the column immediately to its left by 1. * Add the value of the base to the top digit in the current column. * Now, perform the subtraction with the new (larger) top digit. 4. Repeat this process for all columns, moving from right to left. ### Vocabulary * **Base**: The number of unique digits (including zero) used in a positional numeral system. For example, base 10 uses digits 0-9, and base 2 uses 0-1. * **Digit**: A single symbol used to represent a number. For instance, in \( 23_5 \), '2' and '3' are digits. * **Carry-over**: A digit that is transferred from one column to the next higher place value column during an addition operation when the sum exceeds the base. * **Borrowing**: The process in subtraction where value is taken from a higher place value column and added to a lower place value column to allow for subtraction when the top digit is smaller than the bottom digit. ### Examples #### Example 1: Addition in Base 2 Calculate \( 110_2 + 11_2 \) Let's align the numbers and add: ``` 1 1 0_2 + 1 1_2 --------- ``` 1. **Rightmost column (units place)**: \( 0 + 1 = 1 \). (1 is less than base 2, so write 1) 2. **Middle column (2s place)**: \( 1 + 1 = 2 \). (2 is equal to base 2, so divide \( 2 \div 2 = 1 \) remainder \( 0 \)). Write down 0, carry over 1. 3. **Leftmost column (4s place)**: \( 1 \) (from the top number) \( + 1 \) (carried over) \( = 2 \). (2 is equal to base 2, so divide \( 2 \div 2 = 1 \) remainder \( 0 \)). Write down 0, carry over 1. 4. **New column (8s place)**: Write down the carried over 1. Therefore, \( 110_2 + 11_2 = 1001_2 \). #### Example 2: Subtraction in Base 6 Calculate \( 34_6 - 15_6 \) Let's align the numbers and subtract: ``` 3 4_6 - 1 5_6 ------- ``` 1. **Rightmost column (units place)**: We need to subtract 5 from 4. Since \( 4 < 5 \), we must borrow from the '3' in the next column. 2. The '3' becomes '2'. The borrowed value is the base, which is 6. So, we add 6 to 4, making the top digit \( 4 + 6 = 10 \). Now subtract: \( 10 - 5 = 5 \). Write down 5. 3. **Leftmost column (6s place)**: We now have '2' (from the original '3' after borrowing) minus '1'. So, \( 2 - 1 = 1 \). Write down 1. Therefore, \( 34_6 - 15_6 = 15_6 \). ### Exercises 1. Add the following numbers in their respective bases: * \( 101_2 + 11_2 \) * \( 23_4 + 13_4 \) * \( 45_7 + 26_7 \) 2. Subtract the following numbers in their respective bases: * \( 110_2 - 11_2 \) * \( 31_5 - 14_5 \) * \( 72_8 - 35_8 \) ### Assignment 1. Perform the following additions: * \( 111_2 + 101_2 \) * \( 43_5 + 24_5 \) * \( 123_4 + 32_4 \) 2. Perform the following subtractions: * \( 1001_2 - 110_2 \) * \( 52_6 - 25_6 \) * \( 103_7 - 45_7 \) ### Summary * **Addition in other bases**: Add digits column by column from right to left. If the sum in a column is equal to or greater than the base, divide the sum by the base. Write the remainder and carry the quotient to the next column. * **Subtraction in other bases**: Subtract digits column by column from right to left. If a top digit is smaller than the bottom digit, you must 'borrow' from the digit to its left. The value borrowed from the higher place value column and added to the current column is equal to the base of the number system. ## Number bases : Multiplication in other bases. ### Introduction Welcome to this lesson on multiplication in other number bases! So far, you've learned about natural numbers and how to represent them in different bases, as well as addition and subtraction in these bases. Today, we'll extend our understanding to include multiplication. Just like with addition and subtraction, the principles of multiplication remain the same, but the way we carry over numbers changes depending on the base we are working in. Understanding this is crucial for mastering number systems beyond our familiar base 10. ### Key Concepts #### Review of Multiplication in Base 10 Before we dive into other bases, let's quickly recall how multiplication works in base 10 (decimal system). When you multiply two numbers like \( 23 \times 4 \), you multiply each digit of 23 by 4, starting from the rightmost digit. * \( 4 \times 3 = 12 \): You write down 2 and carry over 1. * \( 4 \times 2 = 8 \): You add the carried-over 1, making it 9. * The result is 92. This process involves multiplying digits, and whenever the product is 10 or more, we convert it to units and tens, carrying the tens over to the next place value. #### Principle of Multiplication in Other Bases The fundamental principle of multiplication in any base is similar to base 10. You multiply digits normally, but when the product is equal to or greater than the base, you convert it into units and 'base groups' (e.g., fives, eights, twos), and then carry over the 'base groups' to the next position. **Steps for Multiplication in Other Bases:** 1. **Multiply individual digits:** Perform the multiplication of the digits as if they were in base 10. 2. **Convert to the target base:** If the product of any two digits is equal to or greater than the base, convert this base 10 product into the target base. 3. **Record and Carry Over:** Write down the units digit of the converted product and carry over the 'groups' (the left-hand digit(s) of the converted product) to the next place value. 4. **Add Carries:** When multiplying the next set of digits, remember to add any carried-over amounts to the product *before* converting to the target base and carrying again. #### Example: Multiplication in Base 5 Let's multiply \( 23_5 \times 4_5 \). 1. **Multiply the rightmost digit:** \( 3_5 \times 4_5 \) * In base 10, \( 3 \times 4 = 12 \). * Convert 12 to base 5: \( 12 \/ 5 = 2 \) remainder \( 2 \). So, \( 12_{10} = 22_5 \). * Write down 2, carry over 2. 2. **Multiply the next digit:** \( 2_5 \times 4_5 \) * In base 10, \( 2 \times 4 = 8 \). * Add the carried-over 2: \( 8 + 2 = 10 \). * Convert 10 to base 5: \( 10 \/ 5 = 2 \) remainder \( 0 \). So, \( 10_{10} = 20_5 \). * Write down 0, carry over 2. 3. **Record the final carry:** Since there are no more digits to multiply, write down the final carried-over 2. Therefore, \( 23_5 \times 4_5 = 202_5 \). #### Example: Multiplication in Base 2 (Binary) Multiplying in base 2 is simpler because the only digits are 0 and 1. The multiplication table is very basic: * \( 0 \times 0 = 0 \) * \( 0 \times 1 = 0 \) * \( 1 \times 0 = 0 \) * \( 1 \times 1 = 1 \) Let's multiply \( 11_2 \times 10_2 \). ``` 11_2 x 10_2 ----- 00 (11_2 * 0_2) 110 (11_2 * 1_2, shifted one place left) ----- 110_2 (Add the partial products) ``` **Note:** When adding partial products in binary, remember \( 1 + 1 = 10_2 \) (write 0, carry 1), and \( 1 + 1 + 1 = 11_2 \) (write 1, carry 1). ### Vocabulary * **Base:** The number of unique digits, including zero, that a number system uses to represent numbers. (e.g., base 10 uses 0-9, base 2 uses 0-1). * **Binary:** A number system with base 2. * **Carry Over:** The process of moving a group of units from one place value column to the next higher place value column during arithmetic operations. * **Product:** The result of multiplying two or more numbers. * **Partial Product:** An intermediate result in multiplication, especially in long multiplication, obtained by multiplying the multiplicand by one digit of the multiplier. ### Examples #### Example 1: Multiply \( 14_6 \times 3_6 \) 1. Multiply \( 4_6 \times 3_6 \): * In base 10, \( 4 \times 3 = 12 \). * Convert 12 to base 6: \( 12 \/ 6 = 2 \) remainder \( 0 \). So, \( 12_{10} = 20_6 \). * Write down 0, carry over 2. 2. Multiply \( 1_6 \times 3_6 \): * In base 10, \( 1 \times 3 = 3 \). * Add the carried-over 2: \( 3 + 2 = 5 \). * Since 5 is less than the base 6, no further conversion is needed for this digit. * Write down 5. Therefore, \( 14_6 \times 3_6 = 50_6 \). #### Example 2: Multiply \( 21_3 \times 12_3 \) This involves long multiplication: ``` 21_3 x 12_3 ----- ``` **Step 1: Multiply by the rightmost digit of the multiplier (\( 2_3 \))** * \( 1_3 \times 2_3 \): * In base 10, \( 1 \times 2 = 2 \). * Write down 2 (since \( 2 < 3 \)). * \( 2_3 \times 2_3 \): * In base 10, \( 2 \times 2 = 4 \). * Convert 4 to base 3: \( 4 \/ 3 = 1 \) remainder \( 1 \). So, \( 4_{10} = 11_3 \). * Write down 1, carry over 1. * Result of first partial product: \( 112_3 \) (recording the final carry) **Step 2: Multiply by the next digit of the multiplier (\( 1_3 \)), shifted one place left** * \( 1_3 \times 1_3 \): * In base 10, \( 1 \times 1 = 1 \). * Write down 1. * \( 2_3 \times 1_3 \): * In base 10, \( 2 \times 1 = 2 \). * Write down 2. * Result of second partial product: \( 21_3 \) (shifted, so it's \( 210_3 \) effectively) **Step 3: Add the partial products** ``` 112_3 + 210_3 ----- ``` * Rightmost column: \( 2 + 0 = 2 \). * Middle column: \( 1 + 1 = 2 \). * Leftmost column: \( 1 + 2 = 3 \). * In base 3, \( 3_{10} = 10_3 \). * Write down 0, carry over 1. * Final carry: 1. So, \( 21_3 \times 12_3 = 1022_3 \). ### Exercises 1. Multiply the following numbers in the given bases: a. \( 11_2 \times 1_2 \) b. \( 101_2 \times 11_2 \) c. \( 32_4 \times 2_4 \) d. \( 12_4 \times 13_4 \) e. \( 15_8 \times 3_8 \) f. \( 24_6 \times 5_6 \) 2. Convert the numbers to base 10, multiply them, then convert the answer back to the original base to check your work for exercise 1.a and 1.c. ### Assignment 1. Calculate the following products: a. \( 111_2 \times 10_2 \) b. \( 23_5 \times 12_5 \) c. \( 41_7 \times 3_7 \) d. \( 10_3 \times 22_3 \) e. \( 201_4 \times 3_4 \) 2. A binary computer processes data in base 2. If it performs an operation that is equivalent to multiplying \( 1101_2 \) by \( 101_2 \), what will be the result in base 2? ### Summary Today, we explored multiplication in number bases other than base 10. The core idea is to perform multiplication digit by digit, similar to base 10. However, when a product (in base 10) is equal to or greater than the working base, you must convert that product to the working base, write down the units digit, and carry over the 'base groups' to the next position. This method applies consistently to all bases, whether it's binary (base 2), quartenary (base 4), or any other base. Mastering this skill is essential for a deeper understanding of number systems. Keep practicing, and you'll become proficient in multiplying numbers in any base! ## Elements of ℕ and of ℕ* ### Introduction Welcome, Form One students! In this lesson, we will explore some fundamental building blocks of mathematics: the sets of natural numbers. We will learn about two important sets, **N** and **N***, understand what makes them different, and how they are used in our everyday lives. ### Key Concepts #### What are Natural Numbers (ℕ)? Natural numbers are the numbers we use for counting and ordering. They are the non-negative whole numbers. In many parts of the world, including what we will follow in this lesson, the set of natural numbers starts with zero. * The set of natural numbers is denoted by the symbol **ℕ**. * It includes zero and all positive whole numbers. * The elements of ℕ are: 0, 1, 2, 3, 4, 5, ... * We can write this set as: \[ \mathbb{N} = \{0, 1, 2, 3, 4, 5, \dots\} \] #### What is the Set of Non-Zero Natural Numbers (ℕ*)? Sometimes, we need to count things but we cannot have zero. For example, if you are counting the number of students in a classroom, there must be at least one student for the classroom to be 'non-empty'. This is where the set of non-zero natural numbers comes in. * This set is denoted by the symbol **ℕ*** (read as "N star" or "N asterisk"). * It includes all natural numbers **except zero**. * The elements of ℕ* are: 1, 2, 3, 4, 5, ... * We can write this set as: \[ \mathbb{N}^* = \{1, 2, 3, 4, 5, \dots\} \] #### Differences and Relationship between ℕ and ℕ* The main difference between ℕ and ℕ* is the presence of zero. * **ℕ** contains 0. * **ℕ*** does not contain 0. * Every number in ℕ* is also in ℕ. This means ℕ* is a part of ℕ (we say ℕ* is a *subset* of ℕ). * We can express the relationship as: \[ \mathbb{N} = \mathbb{N}^* \cup \{0\} \] (This means ℕ is the combination of ℕ* and the number zero). #### Representing Natural Numbers on a Number Line A number line is a straight line where numbers are marked at equal intervals. It helps us visualize numbers and their order. * To represent natural numbers, we start with 0 at a point, and then mark 1, 2, 3, and so on, moving to the right. * An arrow on the right end indicates that the numbers continue infinitely in that direction. * The points corresponding to natural numbers are usually dots or small vertical lines. ### Vocabulary * **Natural Numbers (ℕ):** The set of non-negative whole numbers used for counting: \( \{0, 1, 2, 3, \dots\} \). * **Set:** A collection of distinct objects or numbers. * **Non-Zero Natural Numbers (ℕ*):** The set of positive whole numbers, excluding zero: \( \{1, 2, 3, \dots\} \). * **Element:** A member of a set. * **Number Line:** A straight line on which numbers are represented as points. ### Examples 1. **Identify if a number belongs to ℕ or ℕ***: * Is 5 an element of ℕ? Yes, because 5 is a non-negative whole number. * Is 0 an element of ℕ*? No, because ℕ* does not include zero. * Is 100 an element of ℕ and ℕ*? Yes, it is in both sets. * Is -3 an element of ℕ? No, because natural numbers are non-negative. * Is 2.5 an element of ℕ? No, because natural numbers are whole numbers. 2. **Listing elements:** * List the first five elements of ℕ: 0, 1, 2, 3, 4. * List the first five elements of ℕ*: 1, 2, 3, 4, 5. 3. **Real-world application:** * The number of eggs in a carton can be represented by a natural number (e.g., 0 eggs, 6 eggs, 12 eggs). So, it belongs to ℕ. * The number of players on a football team (who are actively playing) must be at least 1. If we are counting active players, this would be an element of ℕ*. ### Exercises 1. Fill in the blanks with either ℕ or ℕ*: * a) The number of fingers on one hand belongs to ________. * b) The number of cars in an empty parking lot belongs to ________. * c) The set of numbers \( \{1, 2, 3, 4, \dots\} \) is ________. * d) The set of numbers \( \{0, 1, 2, 3, \dots\} \) is ________. 2. True or False: * a) 0 is an element of ℕ*. * b) Every number in ℕ* is also in ℕ. * c) 7 is an element of both ℕ and ℕ*. * d) -2 is a natural number. 3. List all the natural numbers less than 6. 4. List all the non-zero natural numbers less than 6. ### Assignment 1. Give two examples from real life where you would use numbers from the set ℕ. 2. Give two examples from real life where you would use numbers from the set ℕ*. 3. Consider the number 15. Is it an element of ℕ? Is it an element of ℕ*? Explain your answers. 4. Draw a number line and mark the first 7 elements of ℕ*. 5. Which set is larger, ℕ or ℕ*? Explain your reasoning. ### Summary In this lesson, we learned about two important sets of numbers: * **ℕ (Natural Numbers):** This set includes zero and all positive whole numbers: \( \{0, 1, 2, 3, \dots\} \). * **ℕ* (Non-Zero Natural Numbers):** This set includes all positive whole numbers, excluding zero: \( \{1, 2, 3, \dots\} \). The key difference is that ℕ contains 0, while ℕ* does not. Understanding these sets is crucial for building a strong foundation in mathematics, as they are the basis for many other number systems and operations you will learn in the future. Keep practicing to become familiar with these fundamental concepts! ## Cardinal, Ordinal and Nominal numbers ### Introduction Welcome, Form One students! In mathematics, numbers are very powerful tools. We use them for many different things, not just counting. In this lesson, we will explore three special types of numbers: Cardinal, Ordinal, and Nominal. Each type has a unique job and helps us understand the world around us better. ### Key Concepts #### Cardinal Numbers Cardinal numbers are used to tell us **how many** of something there are. They represent quantity or amount. * **Purpose:** To count items. * **Examples:** * There are **five** apples on the table. * A cat has **four** legs. * Your classroom has **30** students. * **Characteristic:** They answer the question "How many?" #### Ordinal Numbers Ordinal numbers tell us the **position** or **order** of something in a list or sequence. They show rank. * **Purpose:** To indicate position, rank, or sequence. * **Examples:** * She came \( 1^{st} \) in the race. * This is the \( 2^{nd} \) chapter of the book. * He lives on the \( 3^{rd} \) floor. * You are in \( Form \ 1 \). * **Characteristic:** They answer the question "Which one?" or "In what order?" #### Nominal Numbers Nominal numbers are used as **names** or **labels** to identify something. They do not show quantity or order, and you cannot perform mathematical operations (like adding or subtracting) with them in a meaningful way. * **Purpose:** To identify or label things. * **Examples:** * The football player wears jersey number **10**. * My phone number is **0712345678**. * The bus number is **14A**. * Your student ID is **100123**. * **Characteristic:** They act as a name or code. They don't tell you 'how many' or 'which position'. ### Vocabulary * **Cardinal Number:** A number used for counting to indicate quantity. * **Ordinal Number:** A number used to indicate position or order in a sequence. * **Nominal Number:** A number used as a label or name for identification. * **Quantity:** The amount or number of something. * **Position:** The place where someone or something is located. * **Identification:** The process of recognizing or establishing the identity of someone or something. ### Examples Let's look at some sentences and identify the type of number: 1. "There are **seven** days in a week." (Cardinal - tells 'how many' days) 2. "My birthday is on the \( 15^{th} \) of August." (Ordinal - tells the 'position' of the day in the month) 3. "The train number is **T205**." (Nominal - acts as a 'label' for the train) 4. "I live at house number **12B**." (Nominal - acts as a 'label' for the house) 5. "She got the \( 1^{st} \) prize in the competition." (Ordinal - tells her 'position' or rank) 6. "There are **two** cars parked outside." (Cardinal - tells 'how many' cars) ### Exercises Identify whether the underlined number in each sentence is a Cardinal, Ordinal, or Nominal number. 1. My brother is **six** years old. 2. The \( 5^{th} \) student in line gets to lead. 3. The emergency number is **999**. 4. There are **12** months in a year. 5. This is the \( 2^{nd} \) time I've visited this park. 6. My favourite football player is number **7**. ### Assignment 1. Write down three different examples of cardinal numbers you use in your daily life. 2. Think of two situations where you would use an ordinal number to describe position or order. 3. Find two nominal numbers around your home or school (e.g., a bus number, a TV channel number, a product code). 4. Explain the main difference between an ordinal number and a nominal number. ### Summary Today, we learned that numbers have different roles! Cardinal numbers tell us **how many** (quantity), ordinal numbers tell us **which one** (position or order), and nominal numbers act as **names or labels** (identification). Understanding these differences helps us use numbers correctly and understand information better in mathematics and in our everyday lives. ## Whole number powers and operations in the set ℕ ### Introduction Welcome to this lesson on whole number powers and operations! In this lesson, we will explore the fascinating world of natural numbers, understand what powers (or exponents) are, and learn how to perform basic mathematical operations (addition, subtraction, multiplication, and division) correctly using the order of operations. This knowledge is fundamental for all your future mathematics studies. ### Key Concepts #### 1. The Set of Natural Numbers (\( \mathbb{N} \)) The set of natural numbers, denoted by \( \mathbb{N} \), consists of the counting numbers. These are the numbers we use to count objects. * \( \mathbb{N} = \{1, 2, 3, 4, 5, \dots \} \) * Sometimes, the number 0 is also included in the set of natural numbers, but for this lesson, we will consider \( \mathbb{N} \) to start from 1. #### 2. Whole Number Powers (Exponents) Powers are a way of writing repeated multiplication more simply. * **Definition**: A power represents a number multiplied by itself a certain number of times. * **Notation**: If \( a \) is a natural number and \( n \) is a whole number, then \( a^n \) means \( a \) multiplied by itself \( n \) times. * \( a \) is called the **base**. * \( n \) is called the **exponent** or **power**. * The entire expression \( a^n \) is called a **power**. \[ a^n = \underbrace{a \times a \times \dots \times a}_{n \text{ times}} \] * **Examples**: * \( 2^3 \) is read as "2 to the power of 3" or "2 cubed". It means \( 2 \times 2 \times 2 = 8 \). * \( 5^2 \) is read as "5 to the power of 2" or "5 squared". It means \( 5 \times 5 = 25 \). * \( 10^4 \) is read as "10 to the power of 4". It means \( 10 \times 10 \times 10 \times 10 = 10,000 \). * **Special Cases**: * Any natural number raised to the power of 1 is the number itself: \( a^1 = a \). For example, \( 7^1 = 7 \). * Any natural number (except 0) raised to the power of 0 is 1: \( a^0 = 1 \) (for \( a \neq 0 \)). For example, \( 15^0 = 1 \). #### 3. Order of Operations (BODMAS/PEMDAS) To ensure everyone gets the same answer when evaluating an expression with multiple operations, we follow a specific order. This order is commonly remembered using the acronym **BODMAS** or **PEMDAS**. * **B**rackets (or **P**arentheses) * **O**rders (or **E**xponents/Powers) * **D**ivision and **M**ultiplication (from left to right) * **A**ddition and **S**ubtraction (from left to right) #### 4. Operations in the Set \( \mathbb{N} \) ##### a. Addition * **Symbol**: \( + \) * **Closure**: The sum of any two natural numbers is always a natural number. For example, \( 3 + 5 = 8 \), and \( 8 \in \mathbb{N} \). * **Commutative Property**: The order of addition does not change the sum. \( a + b = b + a \). For example, \( 4 + 7 = 7 + 4 = 11 \). * **Associative Property**: The way numbers are grouped in an addition problem does not change the sum. \( (a + b) + c = a + (b + c) \). For example, \( (2 + 3) + 4 = 2 + (3 + 4) = 9 \). ##### b. Subtraction * **Symbol**: \( - \) * **Not Closed in \( \mathbb{N} \)**: The difference between two natural numbers is not always a natural number. For example, \( 3 - 5 = -2 \), which is not in \( \mathbb{N} \). * **Not Commutative**: The order matters. \( a - b \neq b - a \) (unless \( a=b \)). For example, \( 5 - 3 = 2 \), but \( 3 - 5 = -2 \). * **Not Associative**: The grouping matters. \( (a - b) - c \neq a - (b - c) \). ##### c. Multiplication * **Symbol**: \( \times \) or \( \cdot \) * **Closure**: The product of any two natural numbers is always a natural number. For example, \( 6 \times 4 = 24 \), and \( 24 \in \mathbb{N} \). * **Commutative Property**: The order of multiplication does not change the product. \( a \times b = b \times a \). For example, \( 8 \times 2 = 2 \times 8 = 16 \). * **Associative Property**: The way numbers are grouped in a multiplication problem does not change the product. \( (a \times b) \times c = a \times (b \times c) \). For example, \( (2 \times 3) \times 4 = 2 \times (3 \times 4) = 24 \). * **Distributive Property (over addition)**: Multiplication distributes over addition. \( a \times (b + c) = a \times b + a \times c \). For example, \( 3 \times (4 + 5) = 3 \times 4 + 3 \times 5 = 12 + 15 = 27 \). * **Multiplicative Identity**: The number 1 is the multiplicative identity because any number multiplied by 1 remains unchanged. \( a \times 1 = a \). For example, \( 12 \times 1 = 12 \). ##### d. Division * **Symbol**: \( \div \) or \( / \) * **Not Closed in \( \mathbb{N} \)**: The quotient of two natural numbers is not always a natural number. For example, \( 7 \div 2 = 3.5 \), which is not in \( \mathbb{N} \). * **Not Commutative**: The order matters. \( a \div b \neq b \div a \) (unless \( a=b \)). For example, \( 10 \div 5 = 2 \), but \( 5 \div 10 = 0.5 \). * **Not Associative**: The grouping matters. \( (a \div b) \div c \neq a \div (b \div c) \). * **Remainder**: When division is not exact in natural numbers, there is often a remainder. For example, \( 17 \div 3 = 5 \) with a remainder of \( 2 \). ### Vocabulary * **Natural Numbers (\( \mathbb{N} \))**: The counting numbers: \( \{1, 2, 3, \dots \} \). * **Exponent (Power)**: The small number written above and to the right of the base, indicating how many times the base is multiplied by itself. * **Base**: The number being multiplied by itself in a power expression. * **Power**: The result of raising a base to an exponent; also refers to the expression itself (e.g., \( 2^3 \)). * **Squared**: A number raised to the power of 2 (e.g., \( 5^2 \)). * **Cubed**: A number raised to the power of 3 (e.g., \( 2^3 \)). * **Order of Operations**: The specific sequence in which mathematical operations must be performed. * **Commutative Property**: The property that allows the order of operands to be changed without affecting the result (applies to addition and multiplication). * **Associative Property**: The property that allows the grouping of operands to be changed without affecting the result (applies to addition and multiplication). * **Distributive Property**: The property linking multiplication and addition, where multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. * **Closure**: A set is closed under an operation if performing that operation on any two elements of the set always produces an element that is also in the same set. * **Identity Element**: A number that, when combined with another number using a specific operation, leaves the other number unchanged (e.g., 1 for multiplication). ### Examples #### Example 1: Evaluating Powers * Evaluate \( 3^4 \). * \( 3^4 = 3 \times 3 \times 3 \times 3 = 9 \times 3 \times 3 = 27 \times 3 = 81 \) * Evaluate \( 9^0 \). * \( 9^0 = 1 \) #### Example 2: Applying Order of Operations * Evaluate \( 10 + 4 \times 2^3 - 6 \div 3 \). 1. **Powers**: \( 2^3 = 8 \) Expression becomes: \( 10 + 4 \times 8 - 6 \div 3 \) 2. **Multiplication and Division (from left to right)**: * \( 4 \times 8 = 32 \) * \( 6 \div 3 = 2 \) Expression becomes: \( 10 + 32 - 2 \) 3. **Addition and Subtraction (from left to right)**: * \( 10 + 32 = 42 \) * \( 42 - 2 = 40 \) * Therefore, \( 10 + 4 \times 2^3 - 6 \div 3 = 40 \). #### Example 3: Using Brackets * Evaluate \( (15 - 3) \div 2 + 5 \times (1 + 2)^2 \). 1. **Brackets**: First, solve the expressions inside the brackets. * \( (15 - 3) = 12 \) * \( (1 + 2) = 3 \) Expression becomes: \( 12 \div 2 + 5 \times 3^2 \) 2. **Powers**: \( 3^2 = 9 \) Expression becomes: \( 12 \div 2 + 5 \times 9 \) 3. **Multiplication and Division (from left to right)**: * \( 12 \div 2 = 6 \) * \( 5 \times 9 = 45 \) Expression becomes: \( 6 + 45 \) 4. **Addition**: \( 6 + 45 = 51 \) * Therefore, \( (15 - 3) \div 2 + 5 \times (1 + 2)^2 = 51 \). ### Exercises 1. **Evaluate the following powers:** a. \( 4^3 \) b. \( 7^2 \) c. \( 1^5 \) d. \( 12^0 \) e. \( 10^3 \) 2. **Evaluate the following expressions using the order of operations:** a. \( 5 + 3 \times 4 \) b. \( 18 \div 6 - 2 \) c. \( (7 + 5) \times 2 - 10 \) d. \( 2^4 + 3 \times 5 \) e. \( 20 - 2^2 \times 3 \div 6 \) f. \( 8 \times (6 - 3)^2 \div 9 \) 3. **Are the following statements true or false? Justify your answer.** a. \( 5 - 2 = 2 - 5 \) b. \( (3 \times 2) \times 4 = 3 \times (2 \times 4) \) c. \( 10 \div 5 \in \mathbb{N} \) d. \( 2 + 0 \in \mathbb{N} \) (assuming \( 0 \notin \mathbb{N} \)) ### Assignment 1. **Calculate the value of:** a. \( 6^3 - 5^2 + (10 \div 2)^2 \) b. \( 7 \times (4 + 3)^2 - 5 \times 10 \div 2 \) c. \( 100 \div (2^3 + 2) + 3 \times 4^1 \) 2. **Word Problem**: A baker bakes 5 trays of muffins. Each tray has \( 3^2 \) muffins. If she sells 20 muffins, how many muffins are left? 3. **Challenge**: Insert brackets into the expression \( 30 - 5 \times 2 + 8 \div 2 \) to get: a. The largest possible result. b. The smallest possible result. ### Summary In this lesson, we explored the set of natural numbers \( \mathbb{N} = \{1, 2, 3, \dots \} \). We learned that powers (exponents) are a shorthand for repeated multiplication, with \( a^n \) meaning \( a \) multiplied by itself \( n \) times. We also reviewed the four basic operations: addition, subtraction, multiplication, and division, noting their properties within \( \mathbb{N} \). Most importantly, we mastered the **Order of Operations (BODMAS/PEMDAS)**, which is crucial for solving mathematical expressions correctly. Remember to always work through Brackets, Orders, Division/Multiplication (left to right), and then Addition/Subtraction (left to right). ## Number bases :Convert numbers in base 10 to other bases and from one base to another ### Introduction Welcome to this lesson on number bases! In our everyday lives, we commonly use numbers based on a system of ten digits. This is called Base 10. However, numbers can be represented using different systems, each with its own 'base'. Understanding number bases is very important, especially when you think about how computers work. In this lesson, we will learn how to convert numbers from Base 10 to other bases, from other bases back to Base 10, and even between two different non-Base 10 systems. ### Key Concepts #### What is a Number Base? A number base, also called a radix, is the number of unique digits (including zero) that a positional numeral system uses to represent numbers. The base tells us how many units of one order make one unit of the next higher order. For example, in Base 10, ten units make a ten, ten tens make a hundred, and so on. #### Base 10 (Denary System) This is the most common number system we use daily. It uses ten unique digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The place values in Base 10 are powers of 10: * Units place: \( 10^0 = 1 \) * Tens place: \( 10^1 = 10 \) * Hundreds place: \( 10^2 = 100 \) **Example:** The number \( 123_{10} \) means: \[ 123_{10} = (1 \times 10^2) + (2 \times 10^1) + (3 \times 10^0) \] \[ 123_{10} = (1 \times 100) + (2 \times 10) + (3 \times 1) \] \[ 123_{10} = 100 + 20 + 3 = 123 \] #### Other Bases Numbers can be represented in any base. Here are a few examples: ##### Base 2 (Binary System) * Uses only two digits: 0 and 1. * This system is fundamental to how computers store and process information. * Place values are powers of 2 (e.g., units, twos, fours, eights, etc.). **Example:** The number \( 101_2 \) means: \[ 101_2 = (1 \times 2^2) + (0 \times 2^1) + (1 \times 2^0) \] \[ 101_2 = (1 \times 4) + (0 \times 2) + (1 \times 1) \] \[ 101_2 = 4 + 0 + 1 = 5_{10} \] ##### Base 5 * Uses five digits: 0, 1, 2, 3, 4. * Place values are powers of 5. **Example:** The number \( 23_5 \) means: \[ 23_5 = (2 \times 5^1) + (3 \times 5^0) \] \[ 23_5 = (2 \times 5) + (3 \times 1) \] \[ 23_5 = 10 + 3 = 13_{10} \] ##### Base 8 (Octal System) * Uses eight digits: 0, 1, 2, 3, 4, 5, 6, 7. * Place values are powers of 8. **Example:** The number \( 17_8 \) means: \[ 17_8 = (1 \times 8^1) + (7 \times 8^0) \] \[ 17_8 = (1 \times 8) + (7 \times 1) \] \[ 17_8 = 8 + 7 = 15_{10} \] **Important Rule:** In any base \( n \), the digits used must always be less than \( n \). #### Converting from Base 10 to Other Bases (Repeated Division Method) To convert a number from Base 10 to another base, we use the method of **repeated division** by the target base. **Steps:** 1. Divide the Base 10 number by the target base. 2. Record the remainder. 3. Divide the quotient from the previous step by the target base again. 4. Repeat steps 2 and 3 until the quotient becomes 0. 5. Write down the remainders in reverse order (from bottom to top) to get the number in the new base. #### Converting from Other Bases to Base 10 (Expansion Method) To convert a number from any base to Base 10, we use the **expansion method**. **Steps:** 1. Identify the place value of each digit. These are powers of the original base, starting from \( 0 \) for the rightmost digit. 2. Multiply each digit by its corresponding place value. 3. Sum all the products. #### Converting Between Two Non-Base 10 Systems To convert a number from one non-Base 10 system to another non-Base 10 system (e.g., from Base 2 to Base 5), you must use Base 10 as an intermediate step. **Steps:** 1. Convert the number from its original base to Base 10 (using the expansion method). 2. Convert the resulting Base 10 number to the target base (using the repeated division method). ### Vocabulary * **Base (or Radix):** The number of unique digits (including zero) used in a positional numeral system. * **Base 10 (Denary System):** The number system we use daily, with digits 0-9. Each place value is a power of 10. * **Binary System (Base 2):** A number system using only two digits: 0 and 1. Used in computers. * **Octal System (Base 8):** A number system using eight digits: 0-7. * **Decimal:** Another term for Base 10. * **Digit:** A single symbol used to write numbers. * **Place Value:** The value of a digit based on its position in a number. * **Remainder:** The amount left over after a division. * **Quotient:** The result obtained when one number is divided by another. ### Examples #### 1. Convert \( 25_{10} \) to Base 2 Divide 25 by 2 repeatedly: * \( 25 \div 2 = 12 \) remainder \( 1 \) * \( 12 \div 2 = 6 \) remainder \( 0 \) * \( 6 \div 2 = 3 \) remainder \( 0 \) * \( 3 \div 2 = 1 \) remainder \( 1 \) * \( 1 \div 2 = 0 \) remainder \( 1 \) Reading remainders upwards: \( 11001_2 \). So, \( 25_{10} = 11001_2 \). #### 2. Convert \( 13_{10} \) to Base 5 Divide 13 by 5 repeatedly: * \( 13 \div 5 = 2 \) remainder \( 3 \) * \( 2 \div 5 = 0 \) remainder \( 2 \) Reading remainders upwards: \( 23_5 \). So, \( 13_{10} = 23_5 \). #### 3. Convert \( 45_{10} \) to Base 8 Divide 45 by 8 repeatedly: * \( 45 \div 8 = 5 \) remainder \( 5 \) * \( 5 \div 8 = 0 \) remainder \( 5 \) Reading remainders upwards: \( 55_8 \). So, \( 45_{10} = 55_8 \). #### 4. Convert \( 1011_2 \) to Base 10 Expand the number using powers of 2: \[ 1011_2 = (1 \times 2^3) + (0 \times 2^2) + (1 \times 2^1) + (1 \times 2^0) \] \[ = (1 \times 8) + (0 \times 4) + (1 \times 2) + (1 \times 1) \] \[ = 8 + 0 + 2 + 1 = 11_{10} \] So, \( 1011_2 = 11_{10} \). #### 5. Convert \( 34_5 \) to Base 10 Expand the number using powers of 5: \[ 34_5 = (3 \times 5^1) + (4 \times 5^0) \] \[ = (3 \times 5) + (4 \times 1) \] \[ = 15 + 4 = 19_{10} \] So, \( 34_5 = 19_{10} \). #### 6. Convert \( 27_8 \) to Base 10 Expand the number using powers of 8: \[ 27_8 = (2 \times 8^1) + (7 \times 8^0) \] \[ = (2 \times 8) + (7 \times 1) \] \[ = 16 + 7 = 23_{10} \] So, \( 27_8 = 23_{10} \). #### 7. Convert \( 110_2 \) to Base 5 **Step 1: Convert to Base 10** \[ 110_2 = (1 \times 2^2) + (1 \times 2^1) + (0 \times 2^0) \] \[ = (1 \times 4) + (1 \times 2) + (0 \times 1) \] \[ = 4 + 2 + 0 = 6_{10} \] **Step 2: Convert \( 6_{10} \) to Base 5** * \( 6 \div 5 = 1 \) remainder \( 1 \) * \( 1 \div 5 = 0 \) remainder \( 1 \) Reading remainders upwards: \( 11_5 \). So, \( 110_2 = 11_5 \). ### Exercises Try these conversion problems on your own. 1. Convert \( 17_{10} \) to Base 2. 2. Convert \( 31_{10} \) to Base 5. 3. Convert \( 101_2 \) to Base 10. 4. Convert \( 214_5 \) to Base 10. 5. Convert \( 12_8 \) to Base 2. (Remember to go via Base 10!) ### Assignment Practice these questions at home to strengthen your understanding. 1. Convert \( 42_{10} \) to Base 2. 2. Convert \( 67_{10} \) to Base 8. 3. Convert \( 11011_2 \) to Base 10. 4. Convert \( 432_5 \) to Base 10. 5. Convert \( 1011_2 \) to Base 5. 6. Convert \( 31_5 \) to Base 8. ### Summary In this lesson, we explored the fascinating world of number bases. We learned that the familiar Base 10 system is just one way to represent numbers, and other bases like Base 2 (binary), Base 5, and Base 8 are also important. We mastered two key methods: * **Converting from Base 10 to other bases:** Use the repeated division method, recording remainders from bottom to top. * **Converting from other bases to Base 10:** Use the expansion method, multiplying each digit by its place value (a power of the base) and summing the results. Finally, we understood that converting between two non-Base 10 systems always involves Base 10 as an essential intermediate step. Keep practicing these conversions, as they are a fundamental skill in mathematics and computer science!