Tech+ (Episode 5): Notational Systems (1.2), Part 1

Welcome to the fifth episode in our growing Tech+ exam prep series - a dive into the love language of computers: numbers!

Are you scared of numbers? You’re not alone! But I’ve got great news: you don’t need to be a math genius to understand the number systems computers use.

Today, I’ll walk you through Binary, Decimal, Hexadecimal, and Octal—step by step. No rush, no pressure—just clear, simple explanations. Let’s decode the language of machines!

Introduction to Notational Systems

In the world of computing, numbers are everything. But computers don’t use numbers the same way we do—they have their own ways of counting, called notational systems. Today, we’ll explore four of these:

  1. Binary (Base 2)
  2. Decimal (Base 10)
  3. Hexadecimal (Base 16)
  4. Octal (Base 8)

Let’s start with the Decimal system, the one you already know!

1. Decimal System (Base 10)

The Decimal system is the number system we use every day. It’s called ‘Base 10’ because it uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

Each digit in a number has a place, and that place tells us how much the digit is worth.

  • The rightmost place is the ones place (this is where single numbers like 1, 2, or 3 go).
  • The next place to the left is the tens place (where groups of 10 go).
  • The next place to the left is the hundreds place (where groups of 100 go).

Each place is 10 times bigger than the one before it.

Let's look at an example: What does the number 245 mean?

Let’s break it down:

  1. The 2 is in the hundreds place, so it equals 2 × 100 = 200.
  2. The 4 is in the tens place, so it equals 4 × 10 = 40.
  3. The 5 is in the ones place, so it equals 5 × 1 = 5.

Add them all together:

200 + 40 + 5 = 245.

In the Decimal system, the position of each digit shows its value. By combining the values of all the digits, we get the full number.

2. Binary System (Base 2)

The Binary system is the number system computers use, and it’s called ‘Base 2’ because it only has two digits: 0 and 1.

In Binary, each position in a number has a value, just like in the Decimal system. But instead of using powers of 10, Binary uses powers of 2. Don’t worry—it’s easier than it sounds! Each position doubles in value as you move to the left:

  • The rightmost place is the ones place (1).
  • The next place to the left is the twos place (2).
  • The next is the fours place (4).
  • Then comes the eights place (8), and so on.

Each position is either on (1) or off (0), like a light switch. The Binary number is just the sum of all the "on" positions.

Example: What does the Binary number 1011 mean?

Let’s break it down step by step:

  1. Start from the right:
    • The first 1 is in the ones place, so it equals 1 × 1 = 1.
    • The next 1 is in the twos place, so it equals 1 × 2 = 2.
    • The 0 is in the fours place, so it equals 0 × 4 = 0.
    • The last 1 is in the eights place, so it equals 1 × 8 = 8.
  2. Add them together:
    8 + 0 + 2 + 1 = 11.

So, the Binary number 1011 equals 11 in Decimal.

3. Hexadecimal System (Base 16)

The Hexadecimal system, or Base 16, is used to make long Binary numbers shorter and easier to read. It’s called Base 16 because it uses 16 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.

The letters A through F represent numbers higher than 9:

  • A = 10
  • B = 11
  • C = 12
  • D = 13
  • E = 14
  • F = 15

For example and practice, let’s convert the Hexadecimal number 2F into a Decimal number.

First, start with the rightmost digit, F:

  • In Hexadecimal, F = 15.
  • Since it’s in the ones place, multiply it by 1.
  • That gives us 15 × 1 = 15.

Second, move to the left digit, 2:

  • The 2 is in the sixteens place, so multiply it by 16.
  • That gives us 2 × 16 = 32.

Finally, add them together:

  • 32 + 15 = 47.

So, the Hexadecimal number 2F equals 47 in Decimal.

Hexadecimal is just a way to simplify longer Binary numbers, and it’s surprisingly easy to work with once you get the hang of it. Think of each position as representing powers of 16, and just multiply and add—step by step.

Let's try another example, this time with a slightly larger number. Let’s convert the Hexadecimal number 2F4 into a Decimal number.

Start with the rightmost digit, 4:

  • In Hexadecimal, 4 = 4.
  • Since it’s in the ones place, multiply it by 1.
  • 4×1=4

Second, move to the left digit, F:

  • In Hexadecimal, F = 15.
  • The F is in the sixteens place, so multiply it by 16
  • 15×16=240

Third, move to the leftmost digit, 2:

  • The 2 is in the two hundred fifty-sixes place, so multiply it by 256.
  • 2×256=512.

Finally, add them together:

  • 512+240+4=756.

So, the Hexadecimal number 2F4 equals 756 in Decimal.

4. Octal System (Base 8)

The Octal system, or Base 8, is another way to count, and it uses only the digits 0 through 7. It was popular in older programming systems because it helps simplify Binary numbers into smaller, easier-to-read groups.

In Octal, just like in Decimal or Binary, the position of each digit tells us its value. However, instead of powers of 10 or 2, we’re working with powers of 8.

Let's practice by converting the Octal Number '17' to Decimal

Let’s break it down step by step:

  1. Start with the rightmost digit, 7:
    • This digit is in the ones place, so multiply it by 1:
      • 7 × 1 = 7.
  2. Move to the left digit, 1:
    • This digit is in the eights place, so multiply it by 8:
      • 1 × 8 = 8.
  3. Add the results:
    • 8 + 7 = 15.

So, the Octal number 17 equals 15 in Decimal.

Let's do another practice conversion, converting the Octal Number '143' to Decimal. Here’s how to handle a larger Octal number, like 143:

  1. Start with the rightmost digit, 3:
    • It’s in the ones place, so multiply it by 1:
      • 3 × 1 = 3.
  2. Move to the next digit, 4:
    • It’s in the eights place, so multiply it by 8:
      • 4 × 8 = 32.
  3. Move to the leftmost digit, 1:
    • It’s in the sixty-fours place, so multiply it by 64 (because 8 × 8 = 64):
      • 1 × 64 = 64.
  4. Add all the results:
    • 64 + 32 + 3 = 99.

So, the Octal number 143 equals 99 in Decimal.

Octal might seem unfamiliar at first, but it’s just another way to represent numbers. Think of it like counting in groups of 8 instead of 10 or 2. Once you break it down step by step, it’s easy to manage!

Review Questions

Before we wrap up, let’s test your knowledge with a few review questions! Grab a pen and paper if you need to.

  1. What base is the Binary system?
    a) Base 8
    b) Base 10
    c) Base 2*
    d) Base 16
  2. Which digits are used in the Octal system?
    a) 0-9
    b) 0-7*
    c) 0-15
    d) 1-8
  3. The Hexadecimal number '2F' is equivalent to which Decimal number?
    a) 47*
    b) 15
    c) 32
    d) 64
  4. What does the Binary number '1011' convert to in Decimal?
    a) 9
    b) 11*
    c) 13
    d) 15
  5. How many unique digits are in the Hexadecimal system?
    a) 8
    b) 10
    c) 12
    d) 16*
  6. Which power of 2 does the third digit from the right in Binary represent?
    a) 2
    b) 4*
    c) 8
    d) 16
  7. What is the Decimal value of Octal '17'?
    a) 15*
    b) 17
    c) 19
    d) 23
  8. How is the Decimal system also referred to?
    a) Base 8
    b) Base 10*
    c) Base 2
    d) Base 16
  9. What does the Hexadecimal digit 'A' represent in Decimal?
    a) 8
    b) 9
    c) 10*
    d) 11
  10. Why is Hexadecimal used in computing?
    a) It’s easier to read large binary numbers.*
    b) Computers operate in base 16.
    c) It’s used for all programming languages.
    d) It replaces the need for Decimal calculations.

Answer Key

  1. C) Base 2
  2. B) 0-7
  3. A) 47
  4. B) 11
  5. D) 16
  6. B) 4
  7. A) 15
  8. B) Base 10
  9. C) 10
  10. A) It’s easier to read large binary numbers

Closing

Awesome work! If you feel more confident now, hit that like button, subscribe, and share this video with anyone studying for CompTIA Tech+. Be sure to check out my YouTube channel (@k.richardlabbe), The K. Richard Labbe Podcast, and visit my website at krichardlabbe.com for more learning tools. You can also find me on Facebook, Instagram, and X under the name KRichardLabbe (all one word). See you next time!

Thank you so much for watching and supporting this humble project of mine. I'm grateful to have you along on this journey. Let’s keep learning together—one step at a time! Remember: it’s not too late to do something great!

Take me to Tech+ Episode 6

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