Contemporary Mathematics

# 4.3Converting with Base Systems

Contemporary Mathematics4.3 Converting with Base Systems

Figure 4.4 Computers use Base 2, which only uses 0's and 1's, to represent quantity. (credit: modification of work “IMAG0933” by yvanhou/Flickr, CC BY 2.0)

## Learning Objectives

After completing this section, you should be able to:

1. Convert another base to base 10.
2. Write numbers in different base systems.
3. Convert base 10 to other bases.
4. Determine errors in converting between bases.

In our system of numbers, we use base 10, but using base 10 was not a given within other systems. There were other systems that used bases other than 10, as we saw with the Mayans and the Babylonians. The base 10 system comes down to grouping objects in sets of 10, but grouping in sets of 10 only happens if the culture values grouping by that many. We feel 10 is natural because we have 10 fingers. There are other systems using other grouping values, such as 4 or 20.

One good reason for examining other bases is to remind ourselves how we had to learn arithmetic when we were young, memorizing rules for our base 10 system. We had to learn why those arithmetic rules made sense, such as why $1+1=21+1=2$ and $1+2=31+2=3$. Another good reason for learning other base systems is due to computers; their circuitry instead uses base 2.

In this section, we explore other base systems and how to convert between them.

## Conversion of Another Base into Base 10 and Other Bases

We saw in Hindu-Arabic Positional System that our Hindu-Arabic system uses base 10, which is a system using place values of digits that depend on powers of 10 (or, are based on powers of 10). We’ve already worked with bases other than base 10: The Babylonian system was base 60, while the Mayan system was base 20.

To explore how our base 10 system is used, answer the following question: What’s the following quantity: 4,572? You probably said four thousand five hundred seventy-two (no, there is no “and” between hundred and seventy). But why do you think that 4 means four thousand? A very young person when learning their numbers might say that’s a four five seven and two. But you added the context of thousands to the four. Why?

Place value, that’s why. You learned early on that where the numeral was gave it different meanings. Ten thousands, thousands, hundreds, tens, and ones. So, you translate that symbol string (4,572) into “four thousand five hundred seventy-two.” As we saw in Hindu-Arabic Positional System, expanding a Hindu-Arabic number involved writing the number using each digit times its appropriate power of 10. So, we could write 4,572 as $4×103+5×102+7×101+2×100=4×1000+5×100+7×10+2×14×103+5×102+7×101+2×100=4×1000+5×100+7×10+2×1$.

One possible reason we use base 10 is that we have 10 fingers, and in the cultures where the Hindu-Arabic system developed, that became the standard. Other cultures may have used other ways of organizing numbers, perhaps using 20 by including toes, or using 60 because 60 has many divisors. Mathematically though, base 10 is an awkward base to work in since 10 has limited divisors. But we think it is easy and simple because that’s what we’ve been taught to use.

Using a base 10 system means we need 10 symbols to make our numbering system work: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Now imagine that we all only had 6 fingers instead of 10 and our counting system was based on those 6 fingers. We would be counting in groups of 6, not groups of 10. How would this change how we work with quantity?

First, we’d need only six symbols. Let’s use 0, 1, 2, 3, 4, 5. Second, our place values would be based on powers of 6, not powers of 10. For instance, the number 3,024 in base 6 would be $3×63+0×62+2×61+4×603×63+0×62+2×61+4×60$. That is how you can translate a base 6 number into a base 10 number. When we calculate that expression we get $3×63+0×62+2×61+4×60=3×216+0×36+2×6+4×1=648+0+12+4=6643×63+0×62+2×61+4×60=3×216+0×36+2×6+4×1=648+0+12+4=664$.

This means the base 6 number 3,024 is equal to the base 10 number 664.

From now on, if we are using a base 6 number, we will follow it with the subscript 6, like the following: 3,0246 means the number is in base 6.

A base 10 number gets no subscript (it’s the standard). So, 3,024 is a base 10 number. A base 13 number would be 4,67213.

So, a base 6 system uses only the symbols 0, 1, 2, 3, 4, and 5. Also, the place values use powers of 6. However, we still don’t know how to count in base 6. In order to do so, we’d have to know how to represent the quantities larger than five in base 6. Let’s review how our base 10 system works by counting from 0 to 100, which shows how larger values are represented.

In writing the base 10 numbers, you start with these first 10 values:

 0 1 2 3 4 5 6 7 8 9

But you’ve run out of symbols. So, we use two digits:

 10 11 12 13 14 15 16 17 18 19

The 1 out front means you’ve run out of digits one time.

But now you’ve run out twice. Continuing with those numbers gives:

 20 21 22 23 24 25 26 27 28 29

And so on,

 30 31 32 33 34 etc.…

Eventually, you hit the 90s,

 90 91 92 93 94 95 96 97 98 99

And you’ve run out of the digits again! So, we say we’ve run out of digits in the tens place one time, hence:

 100 101 102 103 104 105 106 107 108 109

That’s the pattern we use in base 10. We write out the symbols until we’ve used all the symbols, then add a digit in front that counts how many times we’ve used the digits. Knowing the numbers, or being able to count higher and higher, is necessary to understand how all the arithmetic works, as it all goes back to counting.

The counting pattern is the same for any other base, including base 6. So, let’s start:

 0 1 2 3 4 5

But we’ve run out of symbols! Just like in base 10, we use a second digit, where the first digit will tell us we’ve run out of symbols one time.

 10 11 12 13 14 15

And we use the same pattern:

 20 21 22 23 24 25 30 31 32 33 34 35 40 41 42 43 44 45 50 51 52 53 54 55

But we’ve run out of symbols for that front digit. So, we indicate it the same way as in base 10…by adding a third digit in front, indicating we’ve run out of symbols once in the second place:

 100 101 102 103 104 105 110 111 etc.

The symbol pattern is the same, but truncated. We only use the six symbols. So that is how we represent base 6. Being able to write out these numbers is important when working with addition in the base.

When using a base larger than 10, though, we need more symbols. Instead of creating new symbols, we use capital letters, with A representing the digit for "10," B representing the digit for "11," and so on.

## Example 4.12

### Determining Digits of a Base with Less Than 10 Digits

What are the digits used for base 7?

1.
What are the digits for base 4?

## Example 4.13

### Determining Digits of a Base with More Than 10 Digits

What are the digits used for base 14?

1.
What are the digits used for base 12?

## Who Knew?

### Using Base 12

As mentioned in the text, working in base 10 is mathematically awkward. Ten has only two natural number divisors: 2 and 5. This means dividing into groups is not easy. However, 12, or a dozen, has more divisors: 2, 3, 4, and 6. The Dozenal Society recognizes this more mathematically pleasant detail. It advocates for a switch to using base 12 for numbers. Their argument is based on the divisibility of the number 12. But has there ever been a society that used such a system? The answer is yes. A dialect of the Gwandara language in Nigeria uses the base 12 system. It is unlikely, though, that the Dozenal Society will achieve their goal, as the base 10 system is so entrenched in our society.

## Example 4.14

### Converting from One Base into Another

Convert 3,6017 into base 10.

## Video

1.
Convert 4216 into base 10.

## Example 4.15

### Converting from Base 14 to Base 10

Convert 4B714 into base 10.

1.
Convert A3C14 into base 10.

## Example 4.16

### Converting from Base 12 to Base 10

Convert A1612 into base 10.

1.
Convert 5AB12 into base 10.

## Example 4.17

### Converting from Base 2 to Base 10

Convert 10112 into base 10.

1.
Convert 110112 into base 10.

## Who Knew?

### Before Napoleon

Before Napoleon’s France, which adopted the base 10 system, a modified base 12 system was often used in Europe. Twelve is easily divisible into groups of 2, 3, 4, and 6, which makes it easier to work with. Even our numbering system retains a bit of this. You have likely noticed that we use the words thirteen, fourteen, fifteen, and so on to indicate 10 and 3, 10 and 4, 10 and 5, and so one. Even the 20s reinforce this idea, as in twenty-one, and twenty-two. However, two numbers don’t follow this pattern, namely 11 and 12. If they followed the same rules, they’d be one teen and two teen. We even have a special word for 12; that is, a dozen. However, etymologically speaking, the words eleven and twelve are likely derived by referencing the number 10. These two numbers may date back to the Old English words endleofan and twelf, which can be traced back further to ain lif and twa lif. The word lif here may be the base word for “to leave.” This would suggest ain lif is one left after 10, and twa lif is two left after 10, or, 11 and 12.

## Example 4.18

### Writing Numbers in Base Systems Other Than Base 10

Write the numbers in base 7 up to 1007.

1.
Write the numbers of base 4 up to 1004.

## Example 4.19

### Writing Numbers in Bases with More Than 10 Symbols

Write the numbers in base 14 up to 10014.

1.
Write the numbers of base 12 up to 1004.

Base 2 is important in the digital age, as it is the system used by computers. It is the simplest base to work with, but has the drawback that the numbers in base 2 may use many, many digits. In Addition and Subtraction in Base Systems and Multiplication and Division in Base Systems, we will look at base 2 in each situation.

## Example 4.20

### Writing Numbers in Base 2

Write the numbers in base 2 up to 1002.

1.
Write the numbers in base 3 up to 1003.

## Who Knew?

### Early Hawaiian Numeration System

Before the British arrived in Hawaii, people there used a system that combined two different bases. Objects were initially grouped into collections of four, and a collection of four was referred to as kauna. A person could have two kauna and three “ones” (in Hindu-Arabic, 11). Or they could have eight kauna and one “ones” (In Hindu-Arabic, 33). However, sets of four were grouped in collections of 10. A set of 10 kauna was ka’au. The collections of ka’au were grouped by 10 also. Which meant that 10 ka’au (this is 40 in Hindu-Arabic) would be lau (or 400 in Hindu-Arabic). What this shows is that the Hawaiian culture developed a system that used base 4 combined with base 10.

## Conversion of Base 10 into Another Base

Converting from base 10 into another base uses repeated division, recording the remainder at each step. Then, the number in the new base is the remainder starting from the last remainder found. To be accurate in what we’re saying, we need to remind ourselves of some terminology associated with division. When integers are divided, the one being divided is the dividend, and the one that is dividing the dividend is the divisor. The quotient is the largest natural number that can be multiplied by the divisor where the product is less than the dividend.

When the integer $nn$ is divided by the integer $dd$, $nn$ is called the dividend and $dd$ is the divisor.

To convert a base 10 number $nn$ into base $dd$, we divide $nn$ by $dd$, recording the remainder. Then we divide the quotient from that step by the base $dd$, and record the remainder again. We continue this process until the quotient is 0. Then, the base $dd$ number has digits that start with the last remainder and use each remainder in reverse order.

## Example 4.21

### Converting from Base 10 into a Lower Base

Convert 298 to base 6.

1.
Convert 693 to base 7.

## Example 4.22

### Converting from Base 10 into a Higher Base

Convert 45,134 to base 13.

1.
Convert 9,275 to base 12.

## Example 4.23

### Converting from Base 10 into Base 2

Convert 100 to base 2.

1.
Convert 137 to base 2.

## Converting from Hindu-Arabic Numbers to Mayan Numbers

To convert from a Hindu-Arabic number to a Mayan number involves two distinct processes. First, the number must be converted to base 20, using the process described and demonstrated previously. Next, that base 20 number has to be written using Mayan numerals. For reference, the Mayan numerals and their values are below.

## Example 4.24

### Converting from Base 10 into the Mayan System

Convert the following into Mayan numbers.

1. 51
2. 653

Convert the following into Mayan numbers.
1.
137
2.
2,171

## Who Knew?

### Other Languages, Other Bases

There have been base systems that use bases other than 10. Some bases used were 20, 12, and 27! Visit this site to see more on the languages that used other bases.

## Errors in Converting Between Bases

There are some common errors that are made when converting between bases. Often, it comes down to using an “illegal” symbol in the new base.

## Example 4.25

### Detecting an Illegal Symbol When Converting Between Bases

A base 10 number is converted to base 7 and the result was 20817. Was an error committed? How do you know?

1.
A base 10 number is converted to base 4 and the result was 37024. Was an error committed? How do you know?

When converting from base 10 to another base, an illegal symbol will be used if a mistake was made in the division process used to find the number in the new base. Since the digits are based on the remainders, any remainder that is an illegal symbol would indicate an error.

## Example 4.26

### Detecting an Error in Division When Converting Between Bases

When changing from base 10 to base 8, the division process resulted in the following remainders: 1, 0, 9, 2, 4. Was an error committed? How do you know?

1.
When changing from base 10 to base 6, the division process resulted in the following remainders: 5, 0, 0, 10. Was an error committed? How do you know?

Another possible way to detect an error in converting between bases is to count the number of digits. When converting from a higher base to a lower base, the number of digits cannot get smaller. Similarly, when converting from a lower base to a higher base, the number of digits cannot get bigger. So, if a base 10 number is converted to a base 3 number, the number of digits in the new base 3 numbers cannot be less than the number of digits in the base 10 number. Similarly, if a base 7 number is converted to base 10, the number of digits in the base 10 number cannot be more than the number of digits in the original base 7 number.

## Example 4.27

### Detecting an Error in Number of Digits When Converting Between Bases

A five-digit base 10 number is converted to a base 5 number. The base 5 number has four digits. Was an error committed? How do you know?

1.
A six-digit base 12 number is converted into a base 10 number. The base 10 number has five digits. Was an error committed? How do you know?

14.
In base 25, how many symbols would be necessary?
15.
In base 18, what would the place value of the 4 be in the number 34818 be?
16.
Convert 23045 into base 10.
17.
When counting in base 9, what number would follow 389?
18.
Convert 329 into base 8.
19.
Convert ABC14 into base 10.
20.
How do you know a mistake was made when converting from base 10 to base 4 and the result is 1524?

## Section 4.3 Exercises

1 .
What does it mean when we say a number is written in base 7?
2 .
What does it mean when we say a number is written in base 12?
3 .
How many symbols are there in a base 3 system? What are they?
4 .
How many symbols are there in a base 15 system? What are they?
5 .
List the numbers, up to 100, in the base 5 system.
6 .
List the numbers, up to 100, in the base 2 system.
For the following exercises, convert the number into a base 10 number.
7 .
145
8 .
216
9 .
123
10 .
345
11 .
148
12 .
789
13 .
3B12
14 .
2416
15 .
1012
16 .
4A714
17 .
8049
18 .
1010012
19 .
32236
20 .
14367
21 .
8A0BD15
22 .
1102023
23 .
100A412
24 .
10100001012
For the following exercises, convert the base 10 number into the given base.
25 .
12 into base 4
26 .
25 into base 2
27 .
43 into base 12
28 .
153 into base 5
29 .
203 into base 2
30 .
431 into base 4
31 .
543 into base 12
32 .
1,023 into base 2
33 .
2,876 into base 4
34 .
1,765 into base 5
35 .
1,993 into base 7
36 .
2,000 into base 2
37 .
4,368 into base 12
38 .
12,562 into base 16 (Hint: Base 16 uses the symbols 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.)
39 .
When converting from base 10 to base 6, a student finds the following remainders: 3, 1, 3, 6. How do you know that a mistake was made?
40 .
When converting from base 10 to base 4, a student finds the following remainders: 0, 0, 3, 7, 2. How do you know a mistake was made?
41 .
Suppose a base 12 number is converted into a base 10 number, and one of the digits is A. Was an error committed? How do you know?
42 .
Suppose a base 10 number is converted into a base 5 number and one of the digits is 6. Was an error committed? How do you know?
43 .
Suppose a base 2 number is converted into a base 10 number, and the base 10 number has more digits that the base 2 number. Was an error committed? How do you know?
44 .
Suppose a base 16 number is converted to base 2, and the base 2 number has fewer digits than the base 16 number. Was an error committed? How do you know?
For the following exercises, convert the Hindu-Arabic number into a Mayan number.
45 .
25
46 .
71
47 .
400
48 .
723

The Babylonian system used base 60. To convert from Hindu Arabic numbers into Babylonian numbers, the process for converting from base 10 to a different base would be done first. Then, the results found in the conversion process would be changed to Babylonian numerals. This process is similar to the one for Mayan numbers.

The Babylonian system used base 60. To convert from Hindu-Arabic numbers into Babylonian numbers, the process for converting from base 10 to a different base would be done first. Then, the results found in the conversion process would be changed to Babylonian numerals. This process is similar to the one for Mayan numbers. For the following exercises, convert the Hindu-Arabic number to a Babylonian number.
49 .
67
50 .
135
51 .
781
52 .
10,952
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