Page 128 - 'Blast_Into_Math

Blast into Math! Mathematical perspectives: all aour mase are melong to us

So now,

2
5
6
100 =1 ∗ 2 +1 ∗ 2 +1 ∗ 2 .
4. We have written 100 as a sum of powers of two. But, we’re not quite done: to finish, we
need to take care of the missing powers of 2. When a power of 2, like in this example 2 is
4
not added, this means that its digit is 0. So, we need to write in these digits,

5
6
2
1
0
3
4
100 =1 ∗ 2 +1 ∗ 2 +0 ∗ 2 +0 ∗ 2 +1 ∗ 2 +0 ∗ 2 +0 ∗ 2 .
To write 100 in base 2 we list its digits, starting from the digit corresponding to the highest
power of the base and continuing until the last digit. So, in base 2, we’d write 100 as 1100100.
Let’s do another example and write 100 in base 5. We’ll follow the same steps.

1. What is the largest power of 5 that is not bigger than 100? Since 5 = 125 > 100, the
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largest power of 5 that is not bigger than 100 is

2
25 =5 .

2. Base 2 is special because the only digits in base 2 are 0 and 1. But now we’re in base 5. So,
the next step is to figure out what is the digit that goes with the power 5 . To do this, we
2
2
find the largestx ∈ Z with 0 ≤ x< 5 such that x ∗ 5 ≤ 100. The largest such x is in
this case x =4. And, in fact,

2
100 =4 ∗ 5 .

3. We’re almost done. We just need to fill in the digits for the missing powers of the base. The
digit for 5 is 4. Since 100 =4 ∗ 5 , the digit of 5 is 0, and the digit of 5 is also 0. So,
1
2
0
2
0
2
1
100 =4 ∗ 5 +0 ∗ 5 +0 ∗ 5 .
4. This means we would write 100 in base 5 as 400.

Now let’s write 100 in base 12.

1. What is the largest power of 12 that is not bigger than 100? Since 12 = 144 > 100, the
2
largest power of 12 that is not bigger than 100 is 12 =12 .
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