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1 Converting from one base to another
1.1 Positional systems
Our numeration system, called the Hindu-Arabic system, is based on the number 10. Other cultures have used di§erent bases for their positional systems. As we have seen, the Mayans used a base 20 system. The Babylonians used a base 60 system. Other popular bases for numeration systems are 5, 8, and 12. (Think about why!)
In a positional system, the position of each digit in the number is important. Letís Örst consider our system, a base 10 system. For example, in the number 432, the ì2îrepresents two 1ís. In the number 423, the ì2î represents two 10ís. In the number 234, the ì2î represents two 100ís. In general, if we look at the number
:::abcd:ef gh:::
the position immediately to the left of the decimal point is the 1ís position. So, the digit d represents that there are d 1ís. The next position to the left is the 10ís position. So, the digit c represents that there are c 10ís. The next position to the left is the 100ís position. So, the digit b represents that there are b 100ís. The next position to the left is the 1000ís position. So, the digit a represents that there are a 1000ís. The dots to the left of the a represent that there could be additional digits to the left; continuing from the a; those places would represent 10,000ís, 100,000ís, 1,000,000ís, etc.
Similarly, we can look at the positions starting immediately to the right of the decimal point. That position is the 101 ís position. So, the digit e represents that there are e 101 ís. The next position to the right is the 1001 ís position. So, the digit f represents that there are f 1001 ís. Again, we can continue this process indeÖnitely to the right. The next position is the 10001 ís position, the next is the 100001 ís position, and so on.
Letís do a speciÖc example. Consider the number 3196 : 403 : The 3 represents that there are three 1000ís, meaning the actual value of the 3 is 3000, or 3 1000 : The 1 represents that there is one 100ís, meaning the actual value of the 1 is 100, or 1 100 : Continuing, we see the actual value of the 9 is 90, or 9 10 ; and the actual value of the 6 is 6 ; or 6 1 : To the right of the decimal point, the 4 represent that there are 4 101 ís, meaning the actual value of the 4 is 104 ; or 4 101 : Contiuing, we see the actual value of the 0 is 1000 ; or 0 1001 ; and the actual value of the 3 is 10003 ; or 3 10001 : Putting this information together, we get
Now, remembering our powers of ten, we can rewrite this number as
3196 :403 = 3 103 + 1 102 + 9 101 + 6 100 + 4 10 ^1 + 0 10 ^2 + 3 10 ^3
(Recall a^0 = 1 and (^) a^1 n = a n^ for any number a 6 = 0.) This is called writing 3196.403 in expanded notation.
Example 1 Write the number 50237 : 004 in expanded notation.
50237 :004 = 5 104 + 0 103 + 2 102 + 3 101 + 7 100 + 0 10 ^1 + 0 10 ^2 + 4 10 ^3
In the above example, since zero times any number is zero, we could shorten the expanded notation above to
50237 :004 = 5 104 + 2 102 + 3 101 + 7 100 + 4 10 ^3
Example 2 Write the number 9000403 : 02001 in expanded notation.
9000403 :02001 = 9 106 + 4 102 + 3 100 + 2 10 ^2 + 1 10 ^5
1.2 Converting from another base to base 10
In the explanation above, there is nothing special about each position representing a power of 10 other than that is what we use. In some instances, it is necessary to use other bases besides 10 for a numeration system. For example, base 2 (binary), base 8 (octal), and base 16 (hexadecimal) systems are very important in computer science. There are also vestiges of numeration systems of di§erent bases in languages around the world. For example, the French word for ìeightyîis ìquatre-vingt,îwhich means ìfour twenties.îAdditionally, there are many di§erent Native American and African languages that emply bases such as three, Öve, and twelve. So, when a number is expressed in a base other than 10, how can we, as native ìbase 10î speakers if you will, discern what the value of a number is? We will illustrate this by means of an example.
Example 3 Express 42068 as a base 10 (Hindu-Arabic) numeral.
The subscripted 8 tells us that the given numeral is expressed in base 8 rather than base 10. This means that each position of the numeral represents a power of 8 rather than a power of 10. Just like before, we start at the decimal point and move left. The 6 represents that there are six 1ís (because 1 = 8^0 ), the 0 represents that there are zero 8ís (because 8 = 8^1 ), the 2 represents that there are two 64 ís (because 64 = 8^2 ), and the 4 represents that there are four 512ís (because 512 = 8^3 ). Using expanded notation like we did before, we see
42068 = 4 83 + 2 82 + 0 81 + 6 80
Now, we can perform the indicated computations to get
42068 = 4 512 + 2 64 + 0 8 + 6 1 = 2048 + 128 + 0 + 6 = 2182
Example 6 In a positional system of base 12, we need digits for each of the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11.
Notice that in a base 12, we need a single digit for the numbers 10 and 11. We cannot use 10 and 11 to represent these numbers because 1012 means 1 121 + 0 120 = 12 and (^1112) means 1 121 + 1 120 = 13: So, we need to make choices for our digits. Typically, we use T 12 to represent 10 and E 12 to represent 11. (We use T 12 and E 12 because T is the Örst letter in ten, and E is the Örst letter in eleven.)
Example 7 Convert 3 T 4 E 12 to a base 10 (Hindu-Arabic) numeral.
3 T 4 E 12 = 3 123 + 10 122 + 4 121 + 11 120 = 5184 + 1440 + 48 + 11 = 6683
Example 8 In a positional system of base 16 (hexadecimal), we need digits for each of the numbers 0, 1, 2, ..., 14, 15. We use A 16 to represent 10 , B 16 to represent 11 , C 16 to represent 12 , D 16 to represent 13 , E 16 to represent 14 , and F 16 to represent 15.
Example 9 Convert B 31 F 16 to a base 10 (Hindu-Arabic) numeral.
B 31 F 16 = 11 163 + 3 162 + 1 161 + 15 160 = 45056 + 768 + 16 + 15 = 45855
1.2.2 Converting from base 10 to another base
When converting from another base to base 10, we used expanded notation to rewrite the given numeral, and then we performed the computations to Önd the corresponding base 10 number. We were able to use expanded notation to rewrite the numeral because we knew the value that each digit represented. However, when we convert from base 10 to another base, we have to Ögure out what digit to put in each position. Again, we illustrate the process we need to do by example.
Example 10 Convert 5402 to base 7.
First, we determine what digit to put in the 1ís position. In base 7, each position represents a power of 7. In particular, the Örst position to the left of the decimal point is the 1ís position, and each of the other positions are positive powers of 7. This means that the digits in each of the other positions represents some multiple of 7. So, in order to determine the digit that goes in the 1ís position, we need to divide the given number by 7; the remainder of this division will be the digit that we put in the 1ís position.
7 540 2
771 R 5
Remember that we can Önd the quotient and remainder using the calculator.
5402 7 = 771 : 7142857 771 : 7142857 771 = 0 : 714285714 0 : 714285714 7 = 5
If in the last step you get something like 4 : 999999999 or 5 : 000000001 ; use 5 as the remainder. The slight di§erence between 5 and these answers represents rounding errors that can occur.
So, 5 will be the digit in the 1ís position. Now, we determine what the digit is in the 7ís position. The quotient of 771 represents how many times 7 goes into 5402. However, we cannot put 771 into the 7ís position. Now, every position to the left of the 7ís position is a multiple of 72 : So, if we divide 771 by 7 and determine the remainder, this remainder will be the digit that goes into the 7ís position. This number will represent the part of 5402 that 7 went into once, but not twice.
7 540 2
7771 R 5
110 R 1
Again, we can Önd the quotient and remainder using the calculator.
771 7 = 110 : 1428571 110 : 1428571 110 = 0 : 14285714 0 : 14285714 7 = 1
So, 1 will be the digit in the 7ís position. Now, 110 cannot go in the 7^2 ís position. So, we divide 110 by 7 again to Önd the remainder; this remainder will be the digit that goes in the 72 ís position.
7 5402
7771 R 5
(^7110) R 1
15 R 5
So, 5 will be the digit in the 7^2 ís position. Again, 15 cannot go in the 7^3 ís position. So, we divide 15 by 7 to Önd the remainder; this remainder will be the digit that goes in the 73 ís position.
7 5402
7 771 R 5
(^7) 110 R 1
7 15 R 5
2 R 1
