Binary Sequences

MF: lightly clean up to make the text more concise

In case someone wants to do more with the project one day:

On this page, you will learn more about how information is represented inside the computer as sequences of bits.

In many programming languages there's a convention that every text string ends with a null byte, in which all eight bits are zero, a code that's reserved for this purpose. So our "Welcome..." string might actually occupy 44 bytes.

So far we've been working with small chunks of data, from Boolean values (one bit) to characters (eight bits). But of course some information in your computer or smartphone is much bigger than that. For starters, characters aren't generally used one at a time; they're used in text strings such as "Welcome to the Beauty and Joy of Computing." These 43 characters occupy 43 bytes of computer memory. But the real champion users of space are media files: pictures, sounds (mostly music), and video.

If we could see inside the memory's bits, a section of the memory might look something like this:

01000001001000000111001101101101011000010111001001110100001000000111000001101000011011110110111001100101001011000010000001101100011010010110101101100101001000000111100101101111011101010111001000100000011011000110000101110000011101000110111101110000001000000110111101110010001000000110010001100101011100110110101101110100011011110111000000100000011000110110111101101101011100000111010101110100011001010111001000101100001000000110001101100001011011100

A binary sequence (also called a bitstream) is a string of ones and zeros.

That shows just 449 bits. A 16GB cell phone has 16 gigabytes (about 16 billion bytes) of storage with each byte containing 8 bits. That's 128,000,000,000 bits. Printed on paper as ones and zeros, the 16GB phone's memory would take nearly 40,000,000 pages. The information in storage—whether it is a text message, a photograph, a song, a computer program, or a list of phone numbers—all looks the same, like a sequence of bits that are either On or Off (one or zero), a binary sequence.

How much information fits in a gigabyte?

Here are a few rough examples of what kind of data would fit in how much memory:

name amount example
bit either a 1 or a 0 1
byte 8 bits 11011001
kilobyte 210 (1,024) bytes a couple of paragraphs
megabyte 220 (1,048,576) bytes about 1 book
gigabyte 230 (1,073,741,824) bytes a little more than 1 CD
terabyte 240 (1,099,511,627,776) bytes about 1,500 CDs
petabyte 250 (1,125,899,906,842,624) bytes about 20 million filing cabinets of text
exabyte 260 (1,152,921,504,606,846,976) bytes about 20% of all the words ever spoken by humankind

As we write this in 2017, it's common to have a terabyte disk drive on your desk. Web services deal with petabytes or exabytes of data.

Where do these prefixes like "tera-" and "peta-" come from?

When we write big numbers, we put commas every three digits (counting from the right). Each group of three has a name: thousand, million, billion, and so on. So, the number 1,234,567,890 is pronounced "one billion, 234 million, 567 thousand, 890." Those group names ("thousand" and so on) also have prefix names used in metric measurements:

prefix amount amount as numeral
kilo- thousand 1,000
mega- million 1,000,000
giga- billion 1,000,000,000
tera- trillion
(a million million)
1,000,000,000,000
peta- quadrillion 1,000,000,000,000,000
exa- quintillion
(a billion billion)
1,000,000,000,000,000,000

Digits for groupings smaller than one (fractions) have metric prefixes too:

prefix amount amount as fraction
milli- thousandth 1/1,000
micro- millionth 1/1,000,000
nano- billionth 1/1,000,000,000
pico- trillionth 1/1,000,000,000,000
femto- quadrillionth 1/1,000,000,000,000,000
atto- quintillionth 1/1,000,000,000,000,000,000
The fractional names are used to measure times in the computer, such as a nanosecond memory access time, or distances between wires on a chip, which are measured in nanometers.
DAT-1.A.6

"Binary sequence" is a very broad category, and often, several layers of abstraction are built on it. For example, you can include a picture in an email or text message, in which case, the message includes a picture, which is a kind of file, which is a kind of binary sequence.

  1. It's unclear what to do here. It might be better to give them some strings to translate. --MF, 6/1/20

    Brian has more or less convinced Mary to change the block names from "convert decimal to binary" to "convert number to binary" and vice versa AND they both agreed to drop the recursion inside the convert number to binary block. The starter project, solutions, and references to it in the page will all need edits. We haven't decided if changes are needed on the Binary Representation page. --MF, 6/5/20

    Click here to load this file. Then save it to your Snap! account.

    Take a look at these 3 custom blocks that you will use to explore binary sequences:

    • A reporter that accepts a string of text as input and translates that text into a binary sequence:
      translate text () to binary sequence
    • A reporter that accepts a binary sequence as input and translates it into text:
      translate binary sequence () to text
    • You can use the second and third inputs to control where the block breaks the sequence to start a new line and also how large the image is drawn.
      A command block that accepts a binary sequence as input and draws a black and white image on the stage where each 0 in the sequence becomes a white "pixel" and each 1 becomes a black "pixel."
      translate binary sequence () to B&W image that is () pixels wide and scaled by factor of ()

  2. Translate a short text string into a binary sequence.
      The quotes tell Snap! that the binary sequence should be saved "as is." Without some text at the beginning, Snap! will think your binary sequence is a number, and it would remove any zeros at the front, which would change the interpretation of where each byte begins and ends.
    • Find the set (output) to... instruction and change the input text to a short text string of your choosing. The reported binary sequence will be stored in the output variable with quotes around it.
      set (output) to (translate text (Beauty and Joy of Computing) to binary sequence)
    • Access the output by right-clicking (or control-clicking) on the OUTPUT watcher on the stage and choosing "export..." The binary sequence will download as a text file. Copy the binary sequence out of the file, but not the quotes.
      image of output watcher on stage storing a binary sequence with the dropdown menu open and  'export...' option selected
  3. Even Snap! has bugs. When you paste this data into Snap!, it may extend beyond the edges of the box. Developers are working to fix this.
    image of output pasted into block and extending past the edges of the block
  4. Paste the outputted binary sequence into the translate binary sequence to text block and run it. (It may take a moment to report.)
    • Is your original text reported back? (If not, you may have included the quotes or lost a bit or two while copying.)
    • Once you've gotten your original text to report back, try making some changes.
      • What happens if you change one bit?
      • What happens if you add a bit somewhere in the middle of the sequence?
      • What happens if you add a bit at the beginning?
  5. DAT-1.A
  6. Go back to the exported output.txt file and copy your original binary sequence again (without the quotes). Paste it into the translate binary sequence to B&W image block and run the block. You are not likely to see anything meaningful. Why not?
  7. Try this binary sequence in translate binary sequence to B&W image with the second input set to 14 pixels wide:

    00000110000000000001000110000000010000000000001100100110000011111111000001100111100000010010110011000111001111100000100110110000000001000000000000110000000000111000000011000100011000010000000100000110000110000000111111000000

    You see should something like the BJC logo:
    Beauty and Joy of Computing logo: a 'b,' a 'j,' and a 'c' looking like two eyes and a nose with a smile below
  8. DAT-1.A
  9. What do you get if you translate that same binary sequence into text? Why?
DAT-1.A.8, DAT-1.A.9, DAT-1.A.10

Analog data have values that change smoothly, unlike digital data which change in discrete intervals.

Sampling means measuring values, called samples, of an analog signal at regular intervals.

The sampling rate is the number of samples measured per second.

Not all data are naturally digital. (That is, they may not be individual values that can be represented in the form of binary sequences.) Some real-world values (such as the pitch and volume of music, the colors of a painting, or the position of a sprinter during a race) change smoothly over time or position; they are analog. When analog data are encoded digitally (as bits on a computer), their values are approximated. This is an example of abstraction. The continuously changing air pressure of a sound, for example, is sampled (measured) thousands of times a second, and the samples are stored as bits.

More examples, please.
A text is a finite number of characters, each of which comes from a finite alphabet. So a book without pictures is naturally digital. When a piece of music is written down, there are a finite number of possible notes, so there's a finite "alphabet" of symbols such as ♫ or 𝄢, and the piece of music uses only finitely many of them. So the score, the written notation for the music, is naturally digital. But the actual sound of words being read or music being played includes much more information (such as they way someone says the words, their accent, or the way each instrument sounds). So speech and musical performance are not naturally digital.

So if pictures, music, and words all look the same in memory—all binary sequences—how can the computer tell what any chunk of memory actually is? For example, should the sequence 01000001 be interpreted as the number 65, the letter A, a rather dark shade of red, or something else?

DAT-1.A.7

The meaning of a sequence of bits depends on the context in which it is used. What exactly do we mean by "context"? How does a programming language know whether to interpret a bit sequence as an integer, a picture, a string of characters, an instruction, or something else? There's always another bit sequence somewhere that encodes the data type of the bit sequence.

AAP-1.A.3 first sentence

But different languages use data types differently. In high-level languages, that data type code is attached to the value itself. In lower-level languages, when you make a variable, you have to say what type of value it will contain, and the data type is attached to the variable, so you can't get exact answers when the values are integers and also be able to handle non-integer values of the same variable. So instead of seeing
script variables (foo)
you see things like
integer (foo)

AAP-1.A.4

Snap! has strengths that many programming languages do not, and it's very likely that your next year's computer science class will use one of those other languages. If that's the case, you'll have to make sure that the data type you declare for a variable matches what you are going to put in it.

  1. This question is similar to those you will see on the AP CSP exam.
    A particular online retail company uses 9-bit binary sequences to identify each unique product for sale. Expecting to increase the number of products it sells, the company is planning to switch to 10-bit binary sequences. Which of the statements below best describes the consequence of using 10-bit sequences instead of 9-bit sequences?
    Two more products can be identified uniquely.
    Ten more products can be identified uniquely.
    Twice as many products can be identified uniquely.
    Ten times as many products can be identified uniquely.
  1. Look inside the translate text to binary sequence and translate binary sequence to text reporters. Describe how these two reporters work. There are several custom blocks inside:
    • pack 8-bit byte takes a binary sequence of 8 bits or less and add enough zeros to the front to make a whole byte. How is this used?
    • translate text to Unicode list takes a text string and outputs a list of each character's Unicode value. Why is a list output helpful here?