118 lines
2.5 KiB
Markdown
118 lines
2.5 KiB
Markdown
---
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tags: [binary, number-systems]
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---
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# Binary number system
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## Decimal (denary) number system
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Binary is a **positional number system**, just like the decimal number system.
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This means that the value of an individual digit is conferred by its position
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relative to other digits. Another way of expressing this is to say that number
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systems work on the basis of **place value**.
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In the decimal system the columns increase by **powers of 10**. This is because
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there are ten total integers in the system:
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$1, 2, 3, 4, 5, 6, 7, 8, 9$
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When we have completed all the possible intervals between $0$ and $9$, we start
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again in a new column.
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The principle of counting in decimal:
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```
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0009
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0010
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0011
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0012
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0013
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...
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0019
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0020
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0021
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```
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Thus each column is ten times larger than the previous column:
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- Ten ($10^1$) is ten times larger than one ($10^0$)
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- A hundred ($10^2$) is ten times larger than ten ($10^1$)
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We use this knowledge of the exponents of the base of 10 to read integers that
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contain multiple digits (i.e. any number greater than 9).
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Thus 264 is the sum of
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$$[4 \cdot (10^0)] + [6 \cdot (10^1)] + [2 \cdot 10^2] $$
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## Binary number system
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In the binary number system, the columns increase by powers of two. This is
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because there are only two integers: 0 and 1. As a result, you are required to
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begin a new column every time you complete an interval from 0 to 1.
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So instead of:
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$$ 10^0, 10^1, 10^2, 10^3 ... (1, 10, 100, 1000) $$
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You have:
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$$ 2^0, 2^1, 2^2, 2^3, 2^4... (0, 2, 4, 8, 16) $$
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When counting in binary, we put zeros as placeholders in the columns we have not
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yet filled. This helps to indicate when we need to begin a new column. Thus the
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counting sequence:
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$$ 1, 2, 3, 4 $$
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is equal to:
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$$ 0001, 0010, 0011, 0100 $$
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Counting in binary:
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```
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000000
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000001
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000010
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000011
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000100
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000101
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000111
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```
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## Binary prefix
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To distinguish numbers in binary from decimal or
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[hexadecimal](Hexadecimal_number_system.md)
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numbers, it is common to use the prefix `0b`. Thus, e.g, `0b110` for decimal
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`6`.
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## Converting decimal to binary
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Let's convert 6 into binary:
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If we have before us the binary place values ($1, 2, 4, 8$). We know that 6 is
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equal to: **1 in the two column and 1 in the 4 column → 110**
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More clearly:
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And for comparison:
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Or we can express the binary as:
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$$ (1 _ 2) + (1 _ 4) $$
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Or more concisely as:
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$$ 2^1 + 2^2 $$
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### Another example
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Let's convert 23 into binary:
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