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 Question from Tammy, a teacher: How can I teach how to change bases? For example 80 base 10 changed to base 3. Thank you!

Hi Tammy,

When I teach change of base I start with a pile of things, maybe toothpicks, coins or other small objects. Suppose I have seventy three of them but I don't tell this to the students. Take these toothpicks and organize them into piles of ten. You get seven piles of size ten and three remaining. Hence there are seven tens and three more or 73 toothpicks. Now imagine repeating the same process with a larger number of toothpicks, say two hundred and seventy three. Put them in piles of ten and you get twenty seven piles and a remainder of three. Take the twenty seven piles of ten and organize them into groups of ten. You get two piles of size one hundred and a remainder of seven. Thus in total you have two plies of size one hundred, seven piles of size ten and three more. Hence you have $273_{10}$ toothpicks.

Now think about what you did arithmetically. I would organize it in a table.

Dividend Quotient Remainder
273 27 3

Put the number you started with in the dividend column, divide by 10 and record the quotient and remainder. Move the quotient into the dividend column and repeat the process.

Dividend Quotient Remainder
273 27 3
27 2 7

Again move the quotient into the dividend column and repeat the process.

Dividend Quotient Remainder
273 27 3
27 2 7
2 0 2

When there is a zero in the quotient column you are finished. The number two hundred and seventy three written in base ten can be read off the remainder column, reading from bottom to top, $273_{10}.$

To write a number in base 3 the process is the same but with the divisor being three rather than ten. Let me illustrate with the seventy three we started with.

Dividend Quotient Remainder
73 24 1

Place the 73 in the dividend column, divide by 3 and record the quotient and remainder. Move the quotient to the dividend column and repeat.

Dividend Quotient Remainder
73 24 1
24 8 0

Move the quotient to the dividend column and repeat.

Dividend Quotient Remainder
73 24 1
24 8 0
8 2 2

Move the quotient to the dividend column and repeat.

Dividend Quotient Remainder
73 24 1
24 8 0
8 2 2
2 0 2

The quotient is zero and you can stop. Reading the remainder column from bottom to top gives $73_{10} = 2201_3$.

I hope this helps,
Harley

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