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The Binary Number System
Robert B. Heckendorn
University of Idaho
August 25, 2015
Numbers are said to be represented by a place value system. Where the value of a symbol depends on where it is... its place. For instance an 8 in right most position in a number means 8 but in the third place from the right means 800.
That is, each position from the right side of the number has an associated value. Base 10 or decimal
Each position to the left is worth 10 more than the next position to the right. For numbers example: in the number 7654, the rightmost place is worth 1, the next to the right most position is worth 10, next position is worth 100 and the left most is worth
1000. The number 7654 is therefore: 7 ∗ 1000 + 6 ∗ 100 + 5 ∗ 10 + 4 ∗ 1 which is 7654 in decimal.
The base 10 number system, numbers are represented by a list of symbols of which there are 10 kinds of symblos: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The position or place values are powers of 10: 100 , 101 , 102 , 103 , ... or 1, 10, 100, 1000, ...
Long ago computing machines used mechanical mechanisms and decimal represention but now with fast electricity based computing devices a new way to encode numbers and data in general is needed.

1

Binary

In binary, the base of the number system is 2.

Base 2 or binary numEach position to the left is worth 2 more than the next position to the right. bers
For example: in the number 1101, the rightmost position is worth 1, the next to the right most position is worth 2, next position is worth 4 and the left most is worth
8. The number 1101 is therefore: 1 ∗ 8 + 1 ∗ 4 + 0 ∗ 2 + 1 ∗ 1 which is 13 in decimal.
In short converting from binary to decimal is as easy as just adding up the binary digits times their place values.
Converting binary to decimal The base 2 number system, numbers are represented by a list of symbols of which there are 2

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