Base Conversion Rules             

Rule 1.   Every number that is spelled out or written in words is assigned the customary value that we give those numbers in our everyday environment. Also, all of the Arabic numerals 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 retain their same conventional decimal system value.   For example, the number five represented by the numeral 5 stands for the same quantity of fingers we have on one hand. 

Rule 2.  The numeral "10" read as "one-zero" now stands for one group of base-many units and zero additional units.  In base seven, the numeral 10 represents one group of seven units.  For example, in our usual base ten monetary systems, 10 pennies has the same value as one dime. But the value of ten pennies is worth two nickels and would be represented by numeral 20 in base five because it represents two nickels or two groups of five pennies.  Only in our more familiar base ten does "one zero," 10 stand for one group of ten units. 

Rule 3.  All of the bases we use are natural numbers greater than or equal to 2 and all exponents are whole numbers greater than or equal 0.  While fractional bases and exponents have corresponding place value meanings as those for the natural numbers we discuss, they are usually not covered in grades K-12.

You are asked to recall that the key to reading anything involving base numbers is to remember the meaning of the numeral one–zero, or the juxtaposition of the digits one and zero. In any base, 10 designates the quantity of one group of base-many and zero units. One group of the base and one unit is denoted as 11 with 12 standing to the base plus two units, etc.

We especially point out that there is no digit for ten in base ten.  The numeral ten is composed of the juxtaposition of the digits one and zero just as with any other base and stands for one group of ten and zero units, 10 .  There are ten numerals in base ten, 0,1,2,3,4,5,6,7,8,9, the last one being nine or one less than ten. Similarly, there are always base-many numerals in any base with the last nonzero numeral corresponding to b-1.  For example, nineteen is one group of ten plus nine units and is written as 19 . But nineteen is written as 12 for one group of twelve plus seven units. So we count from zero to twelve in base twelve, 0, 1, 2, …, 8, 9, T, E, 10 where we need to add base twelve digits standing respectively for ten and eleven units because twelve as the base is denoted by the one-zero notation, 10 .  In base thirteen you also need a digit for twelve.  We choose to use the letter W for that digit and write the numerals 0, 1, 2, … 8, 9, T (ten), E (eleven), W (twelve), 10 (thirteen), 11 (fourteen or thirteen plus one), 12 (fifteen or thirteen plus two), … 1W (twenty-five or one group of thirteen and twelve units) followed by 20 that takes on the value of twenty-six because it represents the equivalent value of two bases and zero units, etc.

Case I.

Base b      Base ten. 

It is easier to convert a number from a base b system to our decimal system than vice versa.  This is because the base b to base ten directions involves only addition and multiplication. You just write the base numeral in expanded base form using the conventional one-zero or "10" place value.  Then replace all the 10's with the appropriate Arabic numeral that has the same value as the base.

      For example, 2643_seven has a value of one thousand eleven units.  We show below how the base seven numeral, 2643_seven is converted to the base ten numeral, 1011_ten.  It doesn't matter which numeral, 2643_seven or 1011_ten we use to represent the value, each stands for the quantity of one thousand eleven units. We first write the number in expanded base seven notation:

This is by first writing out the number in expanded base seven notation as follows:

                                  

Because 10 has the value here of one group of seven units and zero individual units, we just substitute the Arabic numeral seven for every base numera1 10 we see.  This changes only the symbols or numerals used to write out the number.  It does not change the value of the number.

        =   Substitute Arabic 7 for the base numeral "10"           

                                    = 

                                    =               Exponentiation

                                    =           Multiplication

                                    = .                                             Addition

        =                                             Transitivity

It should seem reasonable to you that the base seven numeral, 2643 would need larger place value digits to stand for the same quantity as the base ten numeral 1011 . It takes more boxes to hold one thousand eleven items if one puts only seven things into a single box than if one puts ten things into a single box.

Conversions Between Powers of a Base             Converting from one base to another where the powers of the base numbers are related is easiest if you just make the appropriate substitutions.  For example, any number written in base 9 can be immediately written as a base three number because 3 =9, so 9 =(3 ) .  The following conversion of the quantity eighty-two written using the base nine numeral 101 to a base three numeral shows how straight forward this process can be.  Eighty-two equals nine squared plus zero nines plus one.  NOTE: this example also includes the caveat that you may have to manually insert a zero placeholder in one or more columns of the new base expression as is the case here in the 3 , 3  , and 3 columns.

Example of eighty-two written base nine directly converted to a base three numeral by substituting 32 for 9 and adjusting for the zero placeholders. 101 = 1*9 +0*9 +1*9          =1*(3 ) +0*3 +1*3       =1*3 +0*3 +0*3 +0*3 +1*3       =10001 .

Case II.

Base b     Base ten. 

Some students find it more difficult to convert a number from base ten to another base b than vice versa. This is because the base ten to base b direction involves division as well as addition and multiplication.  We do not have the simplicity in this direction of just writing the number in expanded form and substituting the equivalent Arabic numeral for the base numeral "10".  Instead, this direction involves repeated applications of the Division Algorithm until the remainder when dividing by b for some natural number n is zero.

A reminder of the statement of the division algorithm is appropriate here.

The Division Algorithm applied to the nth step here says: "For every non-negative integer a, b≠0 and whole number n, there exist non-negative integers q and r such that:

                                     if and only if  and .

This algorithm guarantees the existence of a unique non-negative quotient q that is the largest number of times the divisor b can be subtracted from the dividend a leaving a remainder r that satisfies the inequality 0≤r<b .  The caution here is that we need to be sure to account for all cases where the quotient of the b division step is zero. If any quotient is zero, we must remember to put a zero in the matching b^nth column as a placeholder.  Referring to the 3 , 3 , and 3 columns in the example covered in the note above, each 3 divides 1 with a zero quotient and remainder of 1 because 1=0*3 +1. If we omit to enter zero as a placeholder in these columns, we get 11 = 1*3 +1*3    or three plus one which is four and not the correct value eighty-two which is nine squared plus zero nines plus one and is written 10001 .

This shows that when we convert from Base ten to Base b, we much first calculate the values of the different powers of the base, b , b , etc.  Then we must divide the highest nonzero b into the original base ten numbers and continue dividing the next highest power one by one into the remainder of the previous division step.  We stop only when we get to the zero remainder for the units or b column. The quotients of these division steps are the numbers entered into the nth, (n-1), (n-2), etc. columns of the base representation of the number.  To give an example of this process, we take on the challenge of converting 1011 or one thousand eleven to its base seven expression.   We first use the division algorithm for each n (Figure 1.) and then by show how to write out two different sub-algorithms (Figures 2a and 2b) that are often used to shorten the sometimes-tedious division process. 

We first need to determine that 7 =343 is the highest power of seven that will divide into one thousand eleven.  Then we divide 1011 by 343 . We find that 1101=2*343+325 with a quotient of 2 and remainder 325 that is less than the divisor 343. We enter 2 into the cubed place value column and then divide the next highest power of seven, 7 =49 into the remainder from the preceding step.  We do this and get 325=49*6+31; we enter the quotient 6 into the squared column and divide 7 =7 into the remainder 31 getting 31=4*7 +3.  Following the same pattern as before, we enter the quotient 4 into the 7 column and divide the remainder 3 by 7 .  This gives us a zero remainder, 3=3*7 +0.  So, we enter 3 into the 7 or one's column and get the result, 1011 =2643 .  All four of the separate division steps of this conversion are shown in Figure 1.  The quotients, when read from left to right, constitute the base seven numeral for one thousand eleven.

Figure 1

,

Now we can read the values of the quotients from left to right and find that 1101 = 2643

Some students prefer to keep track of the division calculations vertically as is shown in Figure 2a or 2b..  Working vertically, the remainders are read from top to bottom and written horizontally from left to right into their proper base place values

.

                                    Figure 2a                                                    Figure 2b

Before the advent of the calculator, many of us felt we owed a bit of gratitude to those earlier mathematicians who left us the following even easier way to get to the base representation.  Unfortunately, this algorithm gives you the answer without making you aware of the simple mathematical procedures that give it meaning. It requires no mathematical skill other than being able to divide by the single digit number seven.  The proof of this algorithm is written out in all its glory in Figure 3b and is, of course, a direct application of the Division Algorithm and the everyday commutative, associative and distributive properties of our number system.  When you divide 1011 by 7 you get . This gives you a quotient of 144 and a remainder of 3.  Now divide that quotient 144 by 7 and get a remainder of 6.  Your whole number quotients read vertically from top to bottom are written horizontally from the left to right digits and become your new base representation. This same process is called the Euclidean algorithm when it is applied to finding the GCD of two numbers.  We cover the GCD in Chapter 4.

      Figure 3a                                Figure 3b

Simplified Algorithm

Proof:

Fortunately, today you can get the answers even more quickly by using your calculator to do the division steps.  If you do use a calculator, be sure you always put each integer quotient in the right place value column and do not omit any necessary zero placeholders. Just keep dividing the base into the preceding remainder and accounting for your zero quotients until you get a zero remainder.  Individual calculators vary as to what you must do to get a whole number remainder, so consult your calculator's manual for instructions.

            You may use the following templates to practice converting base ten to base b expressions if your wish. 

b4				b4
b3				b3
b2				b2
b1				b1
b0				b0

Course comments:

1.  You are expected to be able to convert from any base b to base ten and vice versa from base ten to any other base b between two and thirteen.  It is also assumed that you are able to clearly explain your work in both directions. 

2.  You will also be assessed on your ability to add, subtract, multiply and divide three digit by two digit numbers in different bases and apply the appropriate properties of our place value number system to explain the how it is the different algorithms we use are valid.

For example,