Number Bases

ACS 2130 - Core Data Structures

Reminder: Self Paced Course Structure

• 1 hour: lecture/activities
• 15 min: Break
• 1.5 hours: Required Lab Time (working on modules)

Modules

• Self assessments
• Mastery portion required
• Stretch are for your own learning if you choose to complete them
• Videos, links, reading, exercises

Learning Outcomes

• Be able to explain the concept of representing numbers in different formats
• Be able to understand the differences between base 10, base 2, and base 16 number systems
• Be able to explain how to convert between various number bases at a high level

Number Bases: There are many ways to represent numbers

Back in the day: Unary Numbers

Basically there’s one symbol which represents 1 and it’s repeated

5: 11111

3:

15:

Eventually number systems evolved to use more than one symbol

Roman: I, II, III, IV, V, X, L, M etc.

Only gave unique symbols to 5, 10, 50, 100, 1000, etc.

Egyptian Numerals

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You can even make your own!

Troll counting system from Terry Pratchett’s Discworld series

one, two, three, many, many-one, many-two, many-three, many many, many-many-one, many-many-two, many-many-three, many many many, many-many-many-one, many-many-many-two, many-many-three, LOTS.

Activity: 10 mins, make your own number system

Think about how would design your own number system

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If you want to use emojis: ctrl + cmd + spacebar brings up the emoji palette

What if we gave each number their own symbol?

🤔

The Hindu-Arabic Numeral System

Use the position of the numeral as well

“We always add and never subtract. And each position is 10 more than the one before it.

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So, 35 means “add 3*10 to 5*1″ and 456 means 4*100 + 5*10 + 6*1. This “positional decimal” setup is the Hindu-Arabic number system we use today.�

Source: https://betterexplained.com/articles/numbers-and-bases/

Base 10 (decimal)

Most modern day cultures use base-10, a.k.a. “decimal.”

We have 10 digits in base-10. (0 through 9)

9 rolls over to 10.

99 rolls over to 100.

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https://www.youtube.com/watch?v=DKavhec9fGE

Base 2 (binary)

Computers use base-2, a.k.a. “binary.”

There are 2 digits in base-2. (0 and 1) 1

1 rolls over to 10.

11 rolls over to 100.

“There are 10 kinds of people in the world. Those who understand binary, and those who don’t.”

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–Old Programming Proverb

Counting in Binary

Binary to Decimal

Let’s Practice Binary to Decimal!

1 0 0 1 0

16 8 4 2 1

16 0 0 2 0

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Flippy Do!

• Paper: https://curriculum.code.org/csp-18/unit1/5/
• Digital: https://docs.google.com/spreadsheets/d/1v4_WCGzdvi6_X5fj_vGu3xt3D3TWGTET5q9yTRJljr0/copy

Decimal to Binary

Remainder Method, keep dividing by 2: https://owlcation.com/stem/How-to-Convert-Decimal-to-Binary-and-Binary-to-Decimal

Let’s Practice Decimal to Binary!

25 / 2 --->

Base 16

Computers sometimes also use base-16, a.k.a. “hexadecimal” or simply “hex.”

There are 16 digits in base-16. (0-9 and A-F)

9 continues to A. F rolls over to 10.

99 continues to 9A. FF rolls over to 100.

Base 16

Base 16

Hex is often prefixed with 0x

0xAF32010016 == AF32010016

Counting in Hex

Hex to Decimal

Check out this article: https://owlcation.com/stem/Convert-Hex-to-Decimal

Post more resources you find on piazza!

Let’s Practice Hex to Decimal!

0x D 3 F 2

(13 × 16³) + (3 × 16²) + (15 × 16¹) + (2 × 16⁰)

0

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How would we go from Decimal to Hex?

Let’s Practice Decimal to Hex!

84 / 16 ---> 5 remainder of 4

5 / 16 r5

54

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Hex to Binary

Let’s Practice Hex to Binary!

0x D 3 F 2

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Think about this...

Based on what we’ve learned how would we convert from decimal to base 4?

Base 4 to decimal?

What about any arbitrary base n?