Number System Lesson: Definition, Types, Conversion and Examples

Lesson Overview

• What Is Number System?
• Types of Number System
• Number Chart
• Number System Conversion

Numbers are fundamental to our understanding of the world, used to quantify, measure, and analyze. But the way we represent these numbers – the symbols and rules we use – can vary significantly. This is where number systems come into play. Each number system is like a unique language with its own  numerical values.

What Is Number System?

A number system is a symbolic framework for representing and manipulating numerical quantities. It provides a structured method for expressing values using a defined set of symbols and rules. These systems are characterized by:

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Types of Number System

Number systems can be broadly categorized based on their base (or radix), which determines the number of unique symbols employed and the place value assigned to each position within a numerical representation.

1. Decimal (Base-10)

Example: Convert the decimal number 345 to its expanded form.

Solution: In the decimal system, each digit's position represents a power of ten.

• 3 is in the hundreds place (10²)

• 4 is in the tens place (10¹)

• 5 is in the units place (10⁰)

• Therefore, 345 = (3 x 10²) + (4 x 10¹) + (5 x 10⁰) = (3 x 100) + (4 x 10) + (5 x 1) = 300 + 40 + 5

2. Binary (Base-2)

• Example: Convert the binary number 10110 to its decimal equivalent.
  

• Solution: In the binary system, each digit's position represents a power of two.

• 1 is in the sixteen's place (2⁴)
• 0 is in the eight's place (2³)
• 1 is in the four's place (2²)
• 1 is in the two's place (2¹)
• 0 is in the units place (2⁰)

• Therefore, 10110 = (1 x 2⁴) + (0 x 2³) + (1 x 2²) + (1 x 2¹) + (0 x 2⁰) = 16 + 0 + 4 + 2 + 0 = 22

3. Octal (Base-8)

• Example: Convert the octal number 237 to its decimal equivalent.

• Solution: In the octal system, each digit's position represents a power of eight.

• 2 is in the sixty-four's place (8²)
• 3 is in the eight's place (8¹)
• 7 is in the units place (8⁰)

• Therefore, 237 = (2 x 8²) + (3 x 8¹) + (7 x 8⁰) = 128 + 24 + 7 = 159

4. Hexadecimal (Base-16)

• Example: Convert the hexadecimal number A3F to its decimal equivalent.
  
• Solution: In the hexadecimal system, each digit's position represents a power of sixteen. Remember A=10, B=11, C=12, D=13, E=14, and F=15.
  
  • A is in the two hundred fifty-six's place (16²)
  • 3 is in the sixteen's place (16¹)
  • F is in the units place (16⁰)

• Therefore, A3F = (10 x 16²) + (3 x 16¹) + (15 x 16⁰) = 2560 + 48 + 15 = 2623

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Number Chart

A number chart is a visual tool that displays numbers in a structured format, often a grid or table. It helps visualize number patterns, sequences, and relationships. Charts can include whole numbers, integers, fractions, or decimals, and are useful for teaching counting, arithmetic, and number sense.

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Number System Shortcut: Quiz!

Number System Conversion

Number system conversion is the process of changing a number's representation from one base to another.  This means taking a number written in one number system (like binary, with base-2) and rewriting it in a different number system (like decimal, with base-10). ²

Converting to Decimal:

• Know your starting point: Figure out the base of the number you're starting with (e.g., binary is base-2, octal is base-8).
• Position is key: Each digit in a number has a place value based on the base. For example, in binary, the rightmost digit is the units place (2⁰), the next is the twos place (2¹), then fours (2²), and so on.
• Multiply and add: Multiply each digit by its place value, then add all the results together.
• Converting from Decimal:
• Divide and conquer: Divide the decimal number by the new base you want to convert to.
• Keep track: Write down the remainder from each division.
• Repeat: Divide the result of the previous division by the new base again. Keep dividing and writing down remainders until you get a result of 0.
• Read from top to button: The remainders you wrote down, read from bottom to top, give you the converted number.

Example 1: Binary to Decimal (1011 to base-10)

• Identify the base: The original number is in binary (base-2).

• Assign place values:
  • The rightmost digit (1) is in the units place (2⁰ = 1).
  • The next digit (1) is in the twos place (2¹ = 2).
  • The next digit (0) is in the fours place (2² = 4).
  • The leftmost digit (1) is in the eights place (2³ = 8).

• Multiply and add:
  • (1 x 8) + (0 x 4) + (1 x 2) + (1 x 1) = 8 + 0 + 2 + 1 = 11
  • Therefore, 1011 (base-2) = 11 (base-10)
    

Example 2: Octal to Decimal (37 to base-10)

• Identify the base: The original number is in octal (base-8).
• Assign place values:
  • The rightmost digit (7) is in the units place (8⁰ = 1).
  • The leftmost digit (3) is in the eights place (8¹ = 8).

Multiply and add:

(3 x 8) + (7 x 1) = 24 + 7 = 31

Therefore, 37 (base-8) = 31 (base-10)

Example 3: Decimal to Binary (13 to base-2)Identify the target base: We want to convert to binary (base-2).

• Divide and keep track of remainders:
  • 13 / 2 = 6 remainder 1
  • 6 / 2 = 3 remainder 0
  • 3 / 2 = 1 remainder 1
  • 1 / 2 = 0 remainder 1

• Read remainders from bottom to top: 1101
  • Therefore, 13 (base-10) = 1101 (base-2)
    

Example 4: Decimal to Hexadecimal (30 to base-16)

• Identify the target base: We want to convert to hexadecimal (base-16).
• Divide and keep track of remainders:
  • 30 / 16 = 1 remainder 14 (which is E in hexadecimal)
  • 1 / 16 = 0 remainder 1

• Read remainders from bottom to top: 1E
  • Therefore, 30 (base-10) = 1E (base-16)
    

Example 5: Hexadecimal to Decimal (2A to base-10)

• Identify the base: The original number is in hexadecimal (base-16).

• Assign place values:
  • The rightmost digit (A, which is 10) is in the units place (16⁰ = 1).
  • The leftmost digit (2) is in the sixteens place (16¹ = 16).
• Multiply and add:
  • (2 x 16) + (10 x 1) = 32 + 10 = 42
  • Therefore, 2A (base-16) = 42 (base-10)

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Can you find out real number system?