Converting from Other Bases to Base [latex]10[/latex]
In the previous sections, we have been referring to positional base systems. In this section of the module, we will explore exactly what a base system is and what it means if a system is “positional.”
A base system is a structure within which we count. The easiest way to describe a base system is to think about our own base-[latex]10[/latex] system. The base-[latex]10[/latex] system, which we call the “decimal” system, requires a total of ten different symbols/digits to write any number. They are, of course, [latex]0[/latex], [latex]1[/latex], [latex]2[/latex], . . . , [latex]9[/latex].
The decimal system is also an example of a positional base system, which simply means that the position of a digit gives its place value. Not all civilizations had a positional system even though they did have a base with which they worked.
base system
Base systems, also known as numeral systems, are ways of representing numbers that rely on a specific ‘base’ or radix to structure the notation.
The most common base system is decimal or base-[latex]10[/latex], which uses ten digits ([latex]0-9[/latex]), but other systems like binary (base-[latex]2[/latex]), octal (base-[latex]8[/latex]), and hexadecimal (base-[latex]16[/latex]) are often used in computing and digital technologies.
Although it is the base our number system uses, base-[latex]10[/latex] is not the only option we have. Practically any positive integer greater than or equal to [latex]2[/latex] can be used as a base for a number system. Such systems can work just like the decimal system except the number of symbols will be different and each position will depend on the base itself.
For example, let’s suppose we adopt a base-five system. The only modern digits we would need for this system are [latex]0[/latex], [latex]1[/latex], [latex]2[/latex], [latex]3[/latex] and [latex]4[/latex]. What are the place values in such a system? To answer that, we start with the ones place, as most base systems do. However, if we were to count in this system, we could only get to four [latex](4)[/latex] before we had to jump up to the next place. Our base is [latex]5[/latex], after all! What is that next place that we would jump to? It would not be tens, since we are no longer in base-[latex]10[/latex]. We’re in a different numerical world. As the base-[latex]10[/latex] system progresses from [latex]10^0[/latex] to [latex]10^1[/latex], so the base-five system moves from [latex]5^0[/latex] to [latex]5^1 = 5[/latex]. Thus, we move from the ones to the fives.
After the fives, we would move to the [latex]5^2[/latex] place, or the twenty-fives. Note that in base-[latex]10[/latex], we would have gone from the tens to the hundreds, which is, of course, [latex]10^2[/latex].
Let’s try an example and build a table.
Consider the number [latex]30412[/latex] in base five. We will write this as [latex]30412_5[/latex], where the subscript [latex]5[/latex] is not part of the number but indicates the base we’re using.
First off, note that this is NOT the number “thirty thousand, four hundred twelve.” We must be careful not to impose the base-[latex]10[/latex] system on this number.
Here’s what our table might look like. We will use it to convert this number to our more familiar base-[latex]10[/latex] system.
| Base-[latex]5[/latex] | This column coverts to Base-[latex]10[/latex] | In Base-[latex]10[/latex] | |
| [latex]3 × 5^4[/latex] | [latex]= 3 × 625[/latex] | [latex]= 1875[/latex] | |
| [latex]+[/latex] | [latex]0 × 5^3[/latex] | [latex]= 0 × 125[/latex] | [latex]= 0[/latex] |
| [latex]+[/latex] | [latex]4 × 5^2[/latex] | [latex]= 4 × 25[/latex] | [latex]= 100[/latex] |
| [latex]+[/latex] | [latex]1 × 5^1[/latex] | [latex]= 1 × 5[/latex] | [latex]= 5[/latex] |
| [latex]+[/latex] | [latex]2 × 5^0[/latex] | [latex]= 2 × 1[/latex] | [latex]= 2[/latex] |
| Total: | [latex]1982[/latex] |
As you can see, the number [latex]30412_5[/latex] is equivalent to [latex]1982[/latex] in base-[latex]10[/latex]. We will say [latex]30412_5 = 1982_{10}[/latex]. All of this may seem strange to you, but that’s only because you are so used to the only system that you’ve probably ever seen.
How to: Convert from Base-[latex]b[/latex] to Base-[latex]10[/latex]
- Identify the place value for each digit in the base-[latex]b[/latex] number, starting from the rightmost place as the “units” place (equivalent to base-[latex]b^0[/latex]), and then increasing the power by one with each position to the left.
- Multiply each digit of the base-[latex]b[/latex] number by the corresponding base-[latex]b[/latex] place value.
- Sum up the values obtained in the second step, which results in the equivalent base-[latex]10[/latex] number.
- Proceed in this way, from right to left, until every digit of the base-[latex]b[/latex] number has been processed.
- The total sum after all operations is your converted base-[latex]10[/latex] number.