Help With Base Numbers?

Question

How can you change the Base 10 Number 17 to Base 4?

What are the place values in Base 4, Base 8, and Base 12.

And can somebody provide a good simple explanation of what base numbers are? Thanks! :)

Answer 1

This is pretty short - you'll need to study it carefully

and probably read more material and practice.

We use base ten, so each place represents a power of 10.

1234 = 1 x 10³ + 2 x 10² + 3 x 10¹ + 4 x 10º

or 1 x 1000 + 2 x 100 + 3 x 10 + 4 x 1

We use digits 0-9 (up to one less than the base).

In other bases the same principle applies.

In base 4, they are powers of 4 (right to left): 1, 4, 16, 64, 256, 1024, 4096, etc

Digits are limited to 0-3. When you hit 4 or more, you go to the next digit.

In base 8, powers of 8: 1, 8, 64, 512, 4096, etc and digits are 0-7.

In base 12, the places are 1, 12, 144, 1728, 20736, etc

Also, in base 12 we need two additional digits for 10 and 11.

T and E are often used for that purpose.

There are two ways to do conversions.

1)

Find the highest power (or multiple of a power), note that down,

subtract and repeat.

17 is too low to be a good example, so let's use 175.

Highest power of 4 in 175 is 64 and there are 2 and a fraction of them.

So we write down 2, note that 2 × 64 = 128, and subtract

175 - 128 = 47

Again there are 2 16's and a fraction (almost 3!),

So we write down another 2, multiply by 16 to get 32,

subtract and we are left with 15.

This time there are 3 and a fraction 4's, so we write down a 3,

multiply by 4 to get 12, subtract leaving 3, which is the last digit.

And we have 2233 (base 4) = 2 x 64 + 2 x 16 + 3 x 4 + 3 x 1

128 + 32 + 12 + 3.

The other way is much simpler, it works in the opposite direction:

Divide the number by 4, note the remainder, repeat using the quotient.

175 div by 4 = 43 remainder 3 (which will be the last digit)

43 div by 4 = 10 remainder 3 (next to last digit)

10 div by 4 = 2 remainder 2 (next digit)

2 div by 4 = 0 remainder 2 (front-most digit)

Number is 2233 (taking the digits in reverse order of generation).

17 in base 4 is 16 + 1 = 101 base 4.

Using the second method:

17 / 4 = 4 remainder 1

4 divide by 4 = 1 remainder 0

1 divide by 4 = 0 remainder 1

Going from bottom up: 101

Here's one more:

110 base 10 into base 4:

Using powers method:

110 between 64 and 256, so note 1 x 64,

leaving 46

46 between 16 and 64, so note 2 x 16

leaving 14

14 between 4 and 16, so note 3 x 4

leaving 2

2 between 1 and 4, so note 2 x 1

leaving 0

Number is 1232

Using remainder method:

110 div by 4 = 27 remainder 2

27 div by 4 = 6 remainder 3

6 div by 4 = 1 remainder 2

1 div by 4 = 0 remainder 1

Number is 1232