**2.1**A variety of abstractions constructed upon binary sequences have the right to be supplied to stand for all digital data.

**2.1.1**describe the range of abstractions supplied to stand for data.

While human being typically work with numbers using the base 10 (decimal) numeral system, other systems are pertinent in computer system science, consisting of binary (base 2) and also hexadecimal (base 16). Computers manage data packed as sequences the bits (**bi**nary digi**ts**), which are all zeros or ones. Human being are most familiar with basic 10, so we write software program that enables people to use base 10 to interact with the computer.

You are watching: The binary number 101 represents the decimal number

In **base 10**, there space ten number (0-9), and each ar is worth *ten* time the location to its right.

In **binary, base 2**, over there are just two digits (0 and also 1), and each ar is precious *two* time the ar to that is right.

The subscript 2 on 11012 method the 1101 is in base 2. Number are normally written in base 10, so a subscript 10 is only supplied when essential for clarity.

clock this Binary Timer Snap

*!*routine run. Create a summary of the binary counter"s behavior. Describe what you see going on.

**Base 2**offers the same idea yet with

**powers of two**instead of strength of ten. Binary ar values represent the units location (20 = 1), the twos place (21 = 2), the fours place (22 = 4), the eights ar (23 = 8), the sixteens ar (24 = 16), etc. So, for example:

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

Here"s a video clip from a various version of smashville247.net. It cut off just before talking around base 16. (You"ll see an ext about reading hexadecimal soon.)

If your connection blocks YouTube, watch the video here, yet you only must watch the very first 3:10—up to (but no including) the component about hexadecimal.

There is a

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

To translate from binary (like 101101_2) to basic 10, **first, write the number out on paper. Then create out the binary location values** by *doubling left indigenous the units place*:

1 | 0 | 1 | 1 | 0 | 1 |

32 | 16 | 8 | 4 | 2 | 1 |

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

To translate from base 10 (like 89_10) to basic 2, **first write out the binary location values** by *doubling left from the systems place* until you obtain to a value bigger than her number (256 because that this example). **Then think**, "My number is smaller sized than 128, for this reason I can leave that place blank. But I have the right to take out a 64, for this reason I compose a 1 there, and also there"s 25 left (89–64). I have actually 0 thirty-twos, because I only have actually 25. However I can take out 16, and there"s 9 left. So, 8 and also 1 space the critical nonzero bits.

Either way you are converting (and between any kind of bases), constantly write the ar values right-to-left (just like with units, tens, hundreds, etc.), and constantly write the number chin left-to-right (just like normal).

89 |

25 |

9 |

1 |

0 |

128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |

1 | 0 | 1 | 1 | 0 | 0 | 1 |

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

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If your link blocks YouTube, clock the video here, however you only must watch the very first 3:14—up come (but no including) the part about hexadecimal.

First, find the biggest power of two that fits within the number. Then, subtract that strength of 2 indigenous the number, save the new number, and record a 1 in the location for that power of 2.

Then, identify if the next largest power of 2 fits within the new number, and: If it does, subtract that power of 2 from the number, keep the new number, and record a 1 in the place for that strength of 2. If it doesn"t, store the exact same number, and record a 0 because that that strength of 2. Repeat this whole step through the next largest power that 2 until you have actually a bit (1 or 0) for all the remaining locations down to and including the ones ar (by which allude you should have nothing left that the original number).

The cable of ones and also zeros you have recorded is the binary representation of your initial number.