Understanding Number Systems
Number systems are fundamental in mathematics, providing a way to represent numbers. In Grade 10, you will encounter various number systems, each with its own rules and applications. Let's explore the key concepts:
1. Decimal Number System
The decimal number system, also known as base-10, is the most common system used in everyday life. It uses ten digits (0-9) to represent numbers. Each digit's position represents a power of 10.
Example: The number 345 can be broken down as:
345 = 3 × 102 + 4 × 101 + 5 × 100
This means 3 is in the hundreds place, 4 is in the tens place, and 5 is in the units place.
2. Binary Number System
The binary number system, or base-2, is used extensively in computer science. It uses only two digits: 0 and 1. Each digit's position represents a power of 2.
Example: The binary number 1101 can be converted to decimal as:
11012 = 1 × 23 + 1 × 22 + 0 × 21 + 1 × 20 = 8 + 4 + 0 + 1 = 1310
3. Octal Number System
The octal number system, or base-8, uses eight digits (0-7). It is less commonly used but still relevant in certain contexts. Each digit's position represents a power of 8.
Example: The octal number 257 can be converted to decimal as:
2578 = 2 × 82 + 5 × 81 + 7 × 80 = 128 + 40 + 7 = 17510
4. Hexadecimal Number System
The hexadecimal number system, or base-16, uses sixteen digits (0-9 and A-F). It is widely used in computer science for its compactness. Each digit's position represents a power of 16.
Example: The hexadecimal number 3A9 can be converted to decimal as:
3A916 = 3 × 162 + 10 × 161 + 9 × 160 = 768 + 160 + 9 = 93710
5. Real Numbers
Real numbers include all the numbers on the number line, including integers, fractions, and irrational numbers. They can be represented in decimal form, which can be either terminating or non-terminating.
Example: The number 2.75 is a terminating decimal, while π (pi) is a non-terminating and non-repeating decimal.
6. Rational and Irrational Numbers
Rational numbers can be expressed as a fraction of two integers, while irrational numbers cannot. Irrational numbers include numbers like π and √2.
Example: 3/4 is a rational number, while √3 is an irrational number.
Understanding these number systems and their properties is crucial for advanced mathematical concepts. Practice converting between different bases and identifying the types of numbers to strengthen your grasp of these concepts.