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### Course: Algebra (all content)>Unit 1

Lesson 14: Binary and hexadecimal number systems

# Introduction to number systems and binary

The base 10 (decimal) system is the most common number system used by humans, but there are other important and useful number systems. For example, base 2, called binary system, is the basis of modern computing. We can convert between the decimal form and binary form of a number to solve different problems.

## Want to join the conversation?

• Why do you not have any punctuation in binary numbers like we do in base-10 numbers (for example: we write 23,567,890,653 but we don't write 111,001,010,111 or 111-001-110-011). I always find myself making mistakes in longer binary numbers (I find this nearly illegible 110110111110110001111110101) and I can't be the only one to lose track, right?
• The reason we put commas every three decimal places has to do with the way we name the value, ... each new comma getting a name. thousands, millions, billions, etc. So we say the number 123,456,789 : one hundred and twenty three million, 4 hundred and fifty six thousand, seven hundred and eighty nine.

In binary, we don't have those names, so commas can't help you say the number. I would argue that every 4 bits should get a space, because many (most?) people that work in binary actually write down (represent) the binary has hexadecimal because it maps cleanly (4bits/hex symbol) so you can write it faster (and use a 1/4 the number of columns)
• In what parts of our life do we use different number bases? Why do we use base 10 for most of the math we do? Also, why is binary useful in computers?
• A computer processor has (nowadays) billions of transistors. Every transistor can be seen as a "switch" that can be on or off (depending on the presence of an electric current). The on state is assigned the value "1", while the off state is assigned the value "0". Each digit is called a "bit" and 8 bits together is a "byte". The binary system was chosen because it was very simple, reliable, and efficient.

In the future we might see quantum computer systems that use all kind of crazy quantum mechanical phenomena.
• I tried doing this with my friend and she gave me the number 715 to translate into binary. However, I didn't find this easy can anyone help me?
• Chloe, you need to understand that to represent any number in binary, you'll need no go up/left to as many places as required to equal or less than the number you wish to represent, just doubling the prior place. So, going up from the 128s place where the video left off we get 256s, 512s, and then 1024s. Since 1024 is more than your example number of 715, we start by placing our most significant (I.e. Largest) bit at the 512s place, subtract 512 from 715 , and you get 203. The next lower place is the 256s, but that is more than our 203 remainder, so we put a 0 in that place and continue. Put the next bit in the 128s place, subtract that amount from 203 and the remainder now is 75. Let's see if you can take it from there?
• Is it true that all numbers can be represented as binary?
• Representing positive integers and zero is pretty straightforward in binary, however, other types of numbers require special rules and handling (that everyone must follow) to represent. These include negative numbers, decimals, fractions, complex (imaginary) numbers, etc.
• How would you do base 12. Like what if we had 6 fingers would it have been base 12 or would it have been decimal?
(1 vote)
• If you just are curious on how to count in base twelve (Duodecimal), then here is an example of how you would count to 100.

1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 2A, 2B, 30,
31, 32, 33, 34, 35, 36, 37, 38, 39, 3A, 3B, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 4A, 4B, 50,
51, 52, 53, 54, 55, 56, 57, 58, 59, 5A, 5B, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 6A, 6B, 70,
71, 72, 73, 74, 75, 76, 77, 78, 79, 7A, 7B, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 8A, 8B, 90,
91, 92, 93, 94, 95, 96, 97, 98, 99, 9A, 9B, A0,
A1, A2, A3, A4, A5, A6, A7, A8, A9, AA, AB, B0,
B1, B2, B3, B4, B5, B6, B7, B8, B9, BA, BB, 100
• Why is there a zero only above 8 and 16?
• The numbers were just digits of the overall number, and he's putting it above the dashes so it shows that they represent the number.
• At , Sal talks about the base-10 system that uses 10 symbols. In view of that, how could we classify the Roman numeral system? It has no 0, but a 10 (X). Also it doesn't have 10 different symbols (I-V-X-L-C-D-M) to express numbers. Is it still considered to be base-10 since it uses multiple of 10s? Thank you for your answers.
• Most sources (but not all) I've seen classify it as base-10, at least in the sense you mention that powers of ten (10, 100, and 1000) are named (X, C, and M); but since the system has no zero and no concept of "place value", it operates differently than positional systems such as decimal and binary.
• what is the binary number system?
• Something that can be hard at first, but you'll eventually understand it. Here is something to use to help : Theoretically, there are an infinite number of them. If you mean how many bases :P
Here are some
Base-2 Binary
Base-3 Ternary
Base-4 Quaternary
Base-5 Quinary
Base-6 Senary
Base-7 Septenary
Base-8 Octal
Base-9 Nonary
Base-10 Decimal
Base-11 Undenary
Base-12 Duodecimal
Base-18 Octodecimal
Base-20 Vigesimal
Base-24 Tetravigesimal
Base-25 Pentavigesimal
Base-30 Trigesimal
Base-32 Duotrigesimal
Base-36 Hexatrigesimal
Base-60 Sexagesimal
Base-64 Tetrasexagesimal
Base-120 Centovigesimal
Base-360 Trecentosexagesimal
I hope this helps!