The Positional System with Other Bases

Learning Outcomes

  • Convert numbers between bases

The Development and Use of Different Number Bases

Other Bases

Powers of numbers other than 10

Recall that in our base-ten number system, each place value in a number represents a power of ten. Our numbers take the form

… thousands hundreds tens ones .

… 103 + 102 + 101 + 100 , where [latex]10^{0}=1[/latex] (In fact any number raised to the zeroth power equals one).

We have an intuitive understanding that [latex]10^{1}=10[/latex] because we are using [latex]10[/latex] as a factor [latex]1[/latex] time. And certainly [latex]10^{2} = 10\times 10 = 100, \\ 10^{3}=1000,[/latex] and so on.

This pattern works with bases other than [latex]10[/latex] as well. As you’ll see below, using [latex]5[/latex] as a base yields the following.

[latex]5^{0}=1 \\ 5^{1}=5 \\ 5^{2}=5\times 5 = 25 \\ 5^{3}=5\times 5\times 5 = 125,[/latex] and so on.

For example, let’s suppose we adopt a base-five system. The only modern digits we would need for this system are 0, 1, 2, 3 and 4. 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 (4) before we had to jump up to the next place. Our base is 5, after all! What is that next place that we would jump to? It would not be tens, since we are no longer in base-ten. We’re in a different numerical world. As the base-ten system progresses from 100 to 101, so the base-five system moves from 50 to 51 = 5. Thus, we move from the ones to the fives.

After the fives, we would move to the 52 place, or the twenty fives. Note that in base-ten, we would have gone from the tens to the hundreds, which is, of course, 102.

Let’s take an example and build a table. Consider the number 30412 in base five. We will write this as 304125, where the subscript 5 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-ten 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-ten system.

Base 5 This column coverts to base-ten In Base-Ten
3 × 54 = 3 × 625 = 1875
+ 0 × 53 = 0 × 125 = 0
+ 4 × 52 = 4 × 25 = 100
+ 1 × 51 = 1 × 5 = 5
+ 2 × 50 = 2 × 1 = 2
Total 1982

As you can see, the number 304125 is equivalent to 1,982 in base-ten. We will say 304125 = 198210. 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.

Example

Convert [latex]6234_{7}[/latex] to a base [latex]10[/latex] number.

Try It

Convert [latex]41065_{7}[/latex] to a base [latex]10[/latex] number.

 

Watch this video to see more examples of converting numbers in bases other than 10 into a base 10 number.

Converting from Base 10 to Other Bases

Converting from an unfamiliar base to the familiar decimal system is not that difficult once you get the hang of it. It’s only a matter of identifying each place and then multiplying each digit by the appropriate power. However, going the other direction can be a little trickier. Suppose you have a base-ten number and you want to convert to base-five. Let’s start with some simple examples before we get to a more complicated one.

Example

Convert twelve to a base-five number.

Example

Convert sixty-nine to a base-four number.

Example

Convert the base-seven number [latex]3261_{7}[/latex] to base 10.

Try It

Convert [latex]143[/latex] to base [latex]5[/latex]

 

Try It

Convert the base-three number [latex]21021_{3}[/latex] to base [latex]10[/latex].

 

In general, when converting from base-ten to some other base, it is often helpful to determine the highest power of the base that will divide into the given number at least once.

In the last example, [latex]5^2= 25[/latex] is the largest power of five that is present in 69, so that was our starting point. If we had moved to [latex]5^3 = 125[/latex], then 125 would not divide into 69 at least once.

Converting from Base 10 to Base b

  1. Find the highest power of the base b that will divide into the given number at least once and then divide.
  2. Write down the whole number part, then use the remainder from division in the next step.
  3. Repeat step two, dividing by the next highest power of the base b, writing down the whole number part (including 0), and using the remainder in the next step.
  4. Continue until the remainder is smaller than the base. This last remainder will be in the “ones” place.
  5. Collect all your whole number parts to get your number in base b notation.

Example

Convert the base-ten number [latex]348[/latex] to base-five.

Solution

The powers of five are:

50 = 1
51 = 5
52 = 25
53 = 125
54 = 625
Etc…

Since 348 is smaller than 625, but bigger than 125, we see that 53 = 125 is the highest power of five present in 348. So we divide 125 into 348 to see how many of them there are:

348 ÷ 125 = 2 with remainder 98

We write down the whole part, 2, and continue with the remainder. There are 98 left over, so we see how many 25s (the next smallest power of five) there are in the remainder:

98 ÷ 25 = 3 with remainder 23

We write down the whole part, 2, and continue with the remainder. There are 23 left over, so we look at the next place, the 5s:

23 ÷ 5 = 4 with remainder 3

This leaves us with 3, which is less than our base, so this number will be in the “ones” place. We are ready to assemble our base-five number:

348 = (2 × 53) + (3 × 52) + (4 × 51) + (3 × 1)

Hence, our base-five number is 2343. We’ll say that [latex]348_{10}=2343_{5}[/latex].

Example

Convert the base-ten number [latex]4,509[/latex] to base-seven.

Solution

The powers of 7 are:

70 = 1
71 = 7
72 = 49
73 = 343
74 = 2401
75 = 16807
Etc…

The highest power of 7 that will divide into 4,509 is 74 = 2401. With division, we see that it will go in 1 time with a remainder of 2108. So we have 1 in the 74 place.

The next power down is 73 = 343, which goes into 2108 six times with a new remainder of 50. So we have 6 in the 73 place.

The next power down is 72 = 49, which goes into 50 once with a new remainder of 1. So there is a 1 in the 72 place.

The next power down is 71 but there was only a remainder of 1, so that means there is a 0 in the 7s place and we still have 1 as a remainder.

That, of course, means that we have 1 in the ones place.

Putting all of this together means that [latex]4,509_{10}=16101_{7}[/latex].

4,509 ÷ 74 = 1 R 21082108 ÷ 73 = 6 R 50

50 ÷ 72 = 1 R 1

1 ÷ 71 = 1

4,50910 = 161017.

Try It

Convert [latex]657_{10}[/latex] to a base [latex]4[/latex] number.

Try It

Convert [latex]8377_{10}[/latex] to a base [latex]8[/latex] number.