Putting It Together: Exponents
Bits and Bytes.
Exponents allow us to perform repeated multiplication quickly, and write very large or small numbers in a way that is easy to understand. There are many and varied uses of exponents that you may be surprised to find are all around us. You may have heard a computer described as having a [latex]32[/latex]-bit processor with [latex]64[/latex] megabytes of RAM and [latex]2.1[/latex] gigabytes of hard disk space. What does that even mean? This terminology is used to describe how fast a computer can process information which results in how quickly it responds after you click on a link or an app or open a file. It also describes how much memory the computer has to store the files and apps you want to use with it.
Computers use binary digits ([latex]0[/latex] and [latex]1[/latex]) instead of the decimal digits ([latex]0[/latex] to [latex]9[/latex]) we are used to working with; therefore, the descriptions [latex]32[/latex]-bit and [latex]64[/latex] megabytes do not make much sense at first glance.
Each binary digit is equivalent to one bit, and a byte contains [latex]8[/latex] bits, often called a binary octet. This is what a byte represents relative to decimal numbers:
Type of Number | Decimal Equivalent | ||||||||
Decimal Powers of 2 | [latex]2^7[/latex] | [latex]2^6[/latex] | [latex]2^5[/latex] | [latex]2^4[/latex] | [latex]2^3[/latex] | [latex]2^2[/latex] | [latex]2^1[/latex] | [latex]2^0[/latex] | |
Decimal Number | [latex]128[/latex] | [latex]64[/latex] | [latex]32[/latex] | [latex]16[/latex] | [latex]8[/latex] | [latex]4[/latex] | [latex]2[/latex] | [latex]1[/latex] | [latex]128+64+32+16+8+4+2+1=255[/latex] |
Binary Octet | [latex]0[/latex] | [latex]0[/latex] | [latex]0[/latex] | [latex]0[/latex] | [latex]0[/latex] | [latex]0[/latex] | [latex]0[/latex] | [latex]1[/latex] | [latex]1[/latex] |
[latex]0[/latex] | [latex]0[/latex] | [latex]0[/latex] | [latex]0[/latex] | [latex]0[/latex] | [latex]0[/latex] | [latex]1[/latex] | [latex]1[/latex] | [latex]2+1=3[/latex] | |
[latex]0[/latex] | [latex]0[/latex] | [latex]0[/latex] | [latex]0[/latex] | [latex]1[/latex] | [latex]1[/latex] | [latex]1[/latex] | [latex]0[/latex] | [latex]8+4+2=14[/latex] | |
[latex]1[/latex] | [latex]0[/latex] | [latex]1[/latex] | [latex]0[/latex] | [latex]1[/latex] | [latex]1[/latex] | [latex]0[/latex] | [latex]0[/latex] | [latex]128+32+8+4=172[/latex] |
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