### Learning Outcomes

- Become familiar with the history of positional number systems
- Identify bases that have been used in number systems historically
- Convert numbers between bases

The Indians were not the first to use a positional system. The Babylonians (as we will see in Chapter 3) used a positional system with 60 as their base. However, there is not much evidence that the Babylonian system had much impact on later numeral systems, except with the Greeks. Also, the Chinese had a base-10 system, probably derived from the use of a counting board.^{[1]} Some believe that the positional system used in India was derived from the Chinese system.

Wherever it may have originated, it appears that around 600 CE, the Indians abandoned the use of symbols for numbers higher than nine and began to use our familiar system where the position of the symbol determines its overall value.^{[2]} Numerous documents from the seventh century demonstrate the use of this positional system.

Interestingly, the earliest dated inscriptions using the system with a symbol for zero come from Cambodia. In 683, the 605th year of the Saka era is written with three digits and a dot in the middle. The 608th year uses three digits with a modern 0 in the middle.^{[3]} The dot as a symbol for zero also appears in a Chinese work (*Chiu**-chih li*). The author of this document gives a strikingly clear description of how the Indian system works:

Using the [Indian] numerals, multiplication and division are carried out. Each numeral is written in one stroke. When a number is counted to ten, it is advanced into the higher place. In each vacant place a dot is always put. Thus the numeral is always denoted in each place. Accordingly there can be no error in determining the place. With the numerals, calculations is easy.^{[4]}

### Transmission to Europe

It is not completely known how the system got transmitted to Europe. Traders and travelers of the Mediterranean coast may have carried it there. It is found in a tenth-century Spanish manuscript and may have been introduced to Spain by the Arabs, who invaded the region in 711 CE and were there until 1492.

In many societies, a division formed between those who used numbers and calculation for practical, every day business and those who used them for ritualistic purposes or for state business.^{[5]} The former might often use older systems while the latter were inclined to use the newer, more elite written numbers. Competition between the two groups arose and continued for quite some time.

In a fourteenth century manuscript of Boethius’ *The Consolations of Philosophy*, there appears a well-known drawing of two mathematicians. One is a merchant and is using an abacus (the “abacist”). The other is a Pythagorean philosopher (the “algorist”) using his “sacred” numbers. They are in a competition that is being judged by the goddess of number. By 1500 CE, however, the newer symbols and system had won out and has persevered until today. The Seattle Times recently reported that the Hindu-Arabic numeral system has been included in the book *The Greatest Inventions of the Past 2000 Years*.^{[6]}

One question to answer is *why* the Indians would develop such a positional notation. Unfortunately, an answer to that question is not currently known. Some suggest that the system has its origins with the Chinese counting boards. These boards were portable and it is thought that Chinese travelers who passed through India took their boards with them and ignited an idea in Indian mathematics.^{[7]} Others, such as G. G. Joseph propose that it is the Indian fascination with very large numbers that drove them to develop a system whereby these kinds of big numbers could easily be written down. In this theory, the system developed entirely within the Indian mathematical framework without considerable influence from other civilizations.

## The Development and Use of Different Number Bases

### Introduction and Basics

During the previous discussions, we have been referring to positional base systems. In this section of the chapter, we will explore exactly what a base system is and what it means if a system is “positional.” We will do so by first looking at our own familiar, base-ten system and then deepen our exploration by looking at other possible base systems. Later in this section, we will journey back to Mayan civilization and look at their unique base system, which is based on the number 20 rather than the number 10.

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-ten system. The base-ten system, which we call the “decimal” system, requires a total of ten different symbols/digits to write any number. They are, of course, 0, 1, 2, . . . , 9.

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.

In our base-ten system, a number like 5,783,216 has meaning to us because we are familiar with the system and its places. As we know, there are six ones, since there is a 6 in the ones place. Likewise, there are seven “hundred thousands,” since the 7 resides in that place. Each digit has a value that is explicitly determined by its position within the number. We make a distinction between digit, which is just a symbol such as 5, and a number, which is made up of one or more digits. We can take this number and assign each of its digits a value. One way to do this is with a table, which follows:

5,000,000 | = 5 × 1,000,000 | = 5 × 10^{6} |
Five million |

+700,000 | = 7 × 100,000 | = 7 × 10^{5} |
Seven hundred thousand |

+80,000 | = 8 × 10,000 | = 8 × 10^{4} |
Eighty thousand |

+3,000 | = 3 × 1000 | = 3 × 10^{3} |
Three thousand |

+200 | = 2 × 100 | = 2 × 10^{2} |
Two hundred |

+10 | = 1 × 10 | = 1 × 10^{1} |
Ten |

+6 | = 6 × 1 | = 6 × 10^{0} |
Six |

5,783,216 | Five million, seven hundred eighty-three thousand, two hundred sixteen |

From the third column in the table we can see that each place is simply a multiple of ten. Of course, this makes sense given that our base is ten. The digits that are multiplying each place simply tell us how many of that place we have. We are restricted to having at most 9 in any one place before we have to “carry” over to the next place. We cannot, for example, have 11 in the hundreds place. Instead, we would carry 1 to the thousands place and retain 1 in the hundreds place. This comes as no surprise to us since we readily see that 11 hundreds is the same as one thousand, one hundred. Carrying is a pretty typical occurrence in a base system.

However, base-ten is not the only option we have. Practically any positive integer greater than or equal to 2 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.

### 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 .

… 10^{3} + 10^{2} + 10^{1} + 10^{0 }, where [latex]10^{0}=1[/latex] (In fact any number raised to the zero^{th }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 10^{0} to 10^{1}, so the base-five system moves from 5^{0} to 5^{1} = 5. Thus, we move from the ones to the fives.

After the fives, we would move to the 5^{2} place, or the twenty fives. Note that in base-ten, we would have gone from the tens to the hundreds, which is, of course, 10^{2}.

Let’s take an example and build a table. Consider the number 30412 in base five. We will write this as 30412_{5}, 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 × 5^{4} |
= 3 × 625 | = 1875 | |

+ | 0 × 5^{3} |
= 0 × 125 | = 0 |

+ | 4 × 5^{2} |
= 4 × 25 | = 100 |

+ | 1 × 5^{1} |
= 1 × 5 | = 5 |

+ | 2 × 5^{0} |
= 2 × 1 | = 2 |

Total | 1982 |

As you can see, the number 30412_{5} is equivalent to 1,982 in base-ten. We will say 30412_{5} = 1982_{10}. 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*

- Find the highest power of the base
*b*that will divide into the given number at least once and then divide. - Write down the whole number part, then use the remainder from division in the next step.
- 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. - Continue until the remainder is smaller than the base. This last remainder will be in the “ones” place.
- 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:

5^{0} = 1

5^{1} = 5

5^{2} = 25

5^{3} = 125

5^{4} = 625

Etc…

Since 348 is smaller than 625, but bigger than 125, we see that 5^{3 }= 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 × 5^{3}) + (3 × 5^{2}) + (4 × 5^{1}) + (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:

7^{0} = 1

7^{1} = 7

7^{2} = 49

7^{3} = 343

7^{4} = 2401

7^{5} = 16807

Etc…

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

The next power down is 7^{3} = 343, which goes into 2108 six times with a new remainder of 50. So we have 6 in the 7^{3} place.

The next power down is 7^{2} = 49, which goes into 50 once with a new remainder of 1. So there is a 1 in the 7^{2} place.

The next power down is 7^{1} 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 ÷ 7^{4} = 1 R 21082108 ÷ 7^{3} = 6 R 50

50 ÷ 7^{2} = 1 R 1

1 ÷ 7^{1} = 1

4,509_{10} = 16101_{7}.

### 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.