In this example,
we're asked to determine the number in base 16
that's equal to 1,260 in base 10.
We know in base 10 the place values
are based upon powers of 10 as shown here.
Where we have 10 to the zero or ones place value.
The 10 to the first or 10s place value.
The 10 to the second or 100s place value.
And the 10 to the thirds or one 1,000s place value.
So the zero goes in the ones place value,
the six is in the 10s place value,
the two is in the 100s place value,
and the one is in the 1,000s place value.
For base 16,
the place values are based upon powers of 16.
Where we have the 16 to the zero or ones place value,
the 16 to the first or 16s place value,
the 16 squared or 256s place value,
and finally the 16 to the third,
or 4,096s place value.
And for base 16 because each place value
can hold the numbers zero through 15,
to avoid the confusion of having a two digit number,
in any given place value,
we use letters to represent the two digit numbers.
So a is equal to 10,
b is equal to 11,
and so on.
Notice how because this place value here
has a value of 4,096,
which is more than 1,260,
the number in base 16 is only going to have three digits.
So starting with the largest place value here,
our first step is to determine
how many 256s are in 1,260.
This would give us a digit in this place value.
So we want to find the quotient 1,260 divided by 256.
If we need to we can use a calculator
to help us with this.
1,260 divided by 256 is given here,
we only want the whole number part of this quotient.
So this tells us there are four 256s in 1,260,
though we are going to have a remainder.
So we have a four here.
Four times 256 is equal to 1,024.
Now we subtract to find the remainder.
This difference is equal to 236.
So if we're doing this by hand,
notice how because the remainder is less than 256,
we know this four is correct.
So because there are four 256s in 1,260,
we know there's going to be a four in this place value,
in base 16.
So now we're going to take the remainder of 236
and determine how many 16s are in 236.
So 236 divided by 16,
the whole number part of the quotient is going to be 14,
but again we can check this on the calculator.
236 divided by 16,
here's our quotient.
Again we're only concerned about
the whole number part of the quotient,
which verifies it is 14.
14 times 16 is equal to 224.
And we subtract to find the remainder.
This difference is 12.
So this quotient tells us we'll have
a 14 in the 16s place value.
Now we're not going to use a 14 in the final number.
We're going to use the letter e.
But we'll leave this as 14 for right now.
And now the last step is to determine
how many ones are in 12.
We know 12 divided by one is equal to 12,
and we have a remainder of zero.
Which means we'll have a 12 in the ones place value.
So we've discovered that 1,260 in base 10
is equal to four times 256,
plus 14 times 16,
plus 12 times one.
So notice how these values here
give us the digits in the place values for base 16.
But again to avoid confusion of having a two digit number
in a given place value,
for 14 we'll use the letter e,
and for 12 we'll use the letter c.
Which means 1,260 base 10 is equal to four e c in base 16.
I hope you found this helpful.