Year 8 Exam>Year 8 Notes>Mathematics for Year 8>Integers
Understanding Integers and Mathematical Operations
Integers
Factors, Multiples, and Primes
- Factors are numbers that divide another number exactly without leaving a remainder. For instance, the factors of 12 include 1, 2, 3, 4, 6, and 12.
- Examples:
- Factors of 20: 1, 2, 4, 5, 10, 20
- Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
- Multiples of a number are obtained by multiplying that number by an integer. For example, the multiples of 5 are 5, 10, 15, 20, and so forth.
- Examples:
- First five multiples of 3: 3, 6, 9, 12, 15
- First five multiples of 7: 7, 14, 21, 28, 35
- Prime numbers are numbers greater than 1 that have no factors other than 1 and themselves. For instance, 2, 3, 5, 7, and 11 are prime numbers.
- Examples:
- Prime numbers between 1 and 20: 2, 3, 5, 7, 11, 13, 17, 19
- Prime numbers between 20 and 40: 23, 29, 31, 37
Multiplying and Dividing Integers
- When multiplying integers, the product of two positive integers results in a positive number, and the product of two negative integers also yields a positive number. However, the product of a positive integer and a negative integer is negative.
| Operation | Result |
|---|---|
| Positive x Positive | Positive |
| Negative x Negative | Positive |
| Positive x Negative | Negative |
Understanding Integer Multiplication and Division
- Integer Multiplication:
- When you multiply two positive integers, the result is positive. For example, 3 * 4 = 12.
- Multiplying two negative integers also yields a positive result. For instance, (-3) * (-4) = 12.
- When you multiply a positive integer by a negative integer, the result is negative. For example, 3 * (-4) = -12.
- Integer Division:
- Division of two positive integers results in a positive quotient. For instance, 12 / 3 = 4.
- Similarly, dividing two negative integers also gives a positive quotient. For example, (-12) / (-3) = 4.
- However, when you divide a positive integer by a negative integer, the quotient is negative. For example, 12 / (-3) = -4.
Understanding Integer Multiplication and Division with Examples
- Examples of Multiplication:
- 3 * 4 = 12
- (-3) * (-4) = 12
- 3 * (-4) = -12
- Examples of Division:
- 12 / 3 = 4
- (-12) / (-3) = 4
- 12 / (-3) = -4
Understanding Division with Integers
Division Basics
Division is the process of splitting a number into equal parts. When dividing integers, it's crucial to understand the rules that apply.
Positive Number Division
When dividing positive numbers, the result is also positive.
- Example: 12 ÷ 4 = 3
Negative Number Division
Dividing negative numbers follows specific rules.
- Example: (-12) ÷ (-4) = 3
Mixed Signs Division
When dividing numbers with mixed signs, the result's sign depends on the numbers involved.
- Example: 12 ÷ (-4) = -3
Understanding Square Roots and Cube Roots
Square Roots
The square root of a number is a value that, when multiplied by itself, equals the original number. For instance, the square root of 16 is 4 because 4 multiplied by 4 gives 16.
Examples:
- √25 = 5
- √49 = 7
Cube Roots
The cube root of a number is a value that, when multiplied by itself three times, results in the original number. For example, the cube root of 27 is 3 because 3 multiplied by 3 multiplied by 3 equals 27.
Mathematical Concepts Explained
- Understanding Multiplication of 3s
- Multiplying three 3s: When you multiply 3 by 3 by 3, you get 27.
- Examples of Cube Roots
- Example 1: Cube root of 8
- Calculating cube root of 8: ∛8 = 2
- Example 2: Cube root of 64
- Calculating cube root of 64: ∛64 = 4
- Example 1: Cube root of 8
Understanding Indices
Indices, also known as exponents or powers, are a way to represent repeated multiplication of a number by itself. For example, 2^3 equals 2 × 2 × 2, which equals 8.
Basic Concepts
- When a number is raised to an exponent, it indicates how many times the base number is multiplied by itself.
- For instance, 3^2 means 3 × 3, which equals 9.
Examples
- 3^2 is the same as 3 × 3, which equals 9.
- 5^3 is equivalent to 5 × 5 × 5, resulting in 125.
Laws of Indices
Understanding certain rules can simplify index calculations:
- Product Rule: When multiplying numbers with the same base, add the exponents. For example, a^m × a^n = a^(m+n).
- Power Rule: When raising an exponent to another exponent, multiply the exponents. For instance, (a^m)^n = a^(m × n).
- Quotient Rule: When dividing numbers with the same base, subtract the exponents. For example, a^m ÷ a^n = a^(m-n).
- Zero Exponent Rule: Any number (except zero) raised to the power of 0 equals 1. For instance, a^0 = 1.
- Multiplication Rule: When multiplying two numbers with the same base, add the exponents. For example, \(a^m \times a^n = a^{m+n}\).
- Example: \(2^3 \times 2^2 = 2^{3+2} = 2^5 = 32\).
- \(a^m \times a^n = a^{m \times n}\).
- Example: \(2^3 \times 2^2 = 2^{3 \times 2} = 2^5 = 32\).
- Division Rule: When dividing two numbers with the same base, subtract the exponents. For example, \(a^m \div a^n = a^{m-n}\).
- Example: \(2^5 \div 2^2 = 2^{5-2} = 2^3 = 8\).
Division Property of Exponents:
- When dividing two powers with the same base, you can subtract the exponents: \(a^m \div a^n = a^{m-n}\)
- Example: \(2^5 \div 2^2 = 2^{5-2} = 2^3 = 8\)
Power of a Power Property:
- When raising a power to another power, you multiply the exponents: \((a^m)^n = a^{mn}\)
- Example: \((2^3)^2 = 2^{3 \times 2} = 2^6 = 64\)
Exponent Rules
Power of a Power
When raising a power to another power, multiply the exponents:
(am)n = amn
For example: (23)2 = 23*2 = 26 = 64
Zero Index Rule
Any non-zero number raised to the power of 0 equals 1:
a0 = 1
For example: 50 = 1
Zero Index:
When a number or variable is raised to the power of 0, it always equals 1.
- Example: \( 5^0 = 1 \)
Negative Index:
When a number or variable is raised to a negative power, it is equivalent to taking the reciprocal of that number or variable raised to the positive power.
- Example: \( 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \)
Understanding and Applying Integer Concepts
- Integers: Integers are whole numbers that can be positive, negative, or zero. They are used in various mathematical operations.
- Factors and Multiples: Factors are numbers that divide a given number without leaving a remainder, while multiples are the result of multiplying a number by an integer.
- Primes: Prime numbers are integers greater than 1 that have no divisors other than 1 and themselves.
- Square Roots and Cube Roots: The square root of a number is a value that, when multiplied by itself, gives the original number. The cube root is similar but involves a number raised to the power of 3.
- Indices: Indices involve exponents or powers, indicating how many times a number is multiplied by itself.
Example Illustration
- Example: $2 - 3 = 1$
$2^3 = 8$
$2^{-3} = \frac{1}{2^3} = \frac{1}{8}$ - By mastering these fundamental concepts, you will be able to solve a wide range of problems related to integers, factors, multiples, primes, square roots, cube roots, and indices effortlessly.
By understanding and applying these concepts, you can handle a variety of problems involving integers, factors, multiples, primes, square roots, cube roots, and indices with ease.
Conclusion
In conclusion, a solid grasp of integer concepts, including factors, multiples, primes, square roots, cube roots, and indices, is essential for tackling mathematical problems effectively and efficiently.
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Document Description: Integers for Year 8 2024 is part of Mathematics for Year 8 preparation. The notes and questions for Integers have been prepared according to the Year 8 exam syllabus. Information about Integers covers topics like Understanding Integers and Mathematical Operations, Understanding Division with Integers and Integers Example, for Year 8 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Integers.
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