2.1 A range of abstractions constructed upon binary sequences have the right to be supplied to stand for all digital data. 2.1.1 describe the variety of abstractions supplied to stand for data. 2.1.1A Digital data is represented by abstractions at different levels. 2.1.1B in ~ the lowest level, every digital data are stood for by bits. 2.1.1C at a higher level, bits space grouped to represent abstractions, including but not minimal to numbers, characters, and color. 2.1.1D Number bases, consisting of binary, decimal, and also hexadecimal, are supplied to represent and also investigate digital data. 2.1.1E At one of the lowest level of abstraction, digital data is represented in binary (base 2) using just combinations that the number zero and also one.

While people typically work with numbers using the base 10 (decimal) numeral system, other systems are pertinent in computer science, consisting of binary (base 2) and also hexadecimal (base 16). Computer systems manage data packed together sequences that bits (binary digits), which room all zeros or ones. Human being are most familiar with base 10, so we write software that permits people to use base 10 to communicate with the computer.

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In base 10, there room ten number (0-9), and also each ar is precious ten times the location to its right.

In binary, base 2, over there are only two digits (0 and also 1), and also each location is worth two time the ar to that right.
The subscript 2 ~ above 11012 means the 1101 is in base 2. Numbers are normally written in base 10, for this reason a subscript 10 is only supplied when essential for clarity.

watch this Binary Timer Snap! routine run. Create a description of the binary counter"s behavior. Define what you watch going on.
Base 2 supplies the exact same idea however with powers of two instead of powers of ten. Binary place values stand for the units location (20 = 1), the twos location (21 = 2), the fours place (22 = 4), the eights ar (23 = 8), the sixteens place (24 = 16), etc. So, for example:

100102 = 1 × 24 + 0 × 23 + 0 × 22 + 1 × 21 + 0 × 20 = 16 + 2 = 1810

Here"s a video from a different version that lennythewonderdog.net. It cut off just prior to talking about base 16. (You"ll see more about reading hexadecimal soon.)

There is a mistake in the video at 2:50. Execute you check out why? (Also, no everyone learns base 10 place values in kindergarten!)

To analyze from binary (like 101101_2) to base 10, first, create the number the end on paper. Then compose out the binary ar values through doubling left indigenous the systems place:

 1 0 1 1 0 1 32 16 8 4 2 1

This means this number is 32 + 8 + 4 + 1. So, 101101_2 = 45_10.

To translate from base 10 (like 89_10) to base 2, first compose out the binary place values by doubling left from the units place till you obtain to a value bigger than your number (256 because that this example). Then think, "My number is smaller than 128, so I have the right to leave that place blank. However I have the right to take the end a 64, so I write a 1 there, and there"s 25 left (89–64). I have actually 0 thirty-twos, because I only have 25. Yet I deserve to take the end 16, and there"s 9 left. So, 8 and 1 room the last nonzero bits.

Either way you are converting (and between any bases), constantly write the ar values right-to-left (just like with units, tens, hundreds, etc.), and always write the number chin left-to-right (just choose normal).
 89 25 9 1 0
 128 64 32 16 8 4 2 1 1 0 1 1 0 0 1

Now, check out the number off: 1011001_2=89_10.

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