Welcome to a lesson on writing numbers of different bases in base 10. For
example in our base 10 system, let's consider the number of 4321. We can
indicate base 10 using this notation here with a subscript of 10. This becomes
important when working with different bases because we're so familiar with the
base 10 system we know this 4 represents 4 1000s
but we can also write 1000 as 10 to the 3rd. The 3 represents three 100's but we
can also write 100 as 10 to the 2nd. The 2 represents 2 tens. We can also write 10
as 10 to the 1st and then finally the 1 represents 1 1, but we can write 1
as 10 to the 0. Notice how we can find the indicated place value by taking the
base and raising it to an integer power. So we could write 4321 as 4 times 10 to
the third plus 3 times 10 squared plus 2 times 10 to the first plus 1 times 10 to
the 0. If we expand this, we do get 4321 in base 10. But now let's adopt a base 5
system the only modern digits we need for base 5 would be 0 1 2 3 & 4. Now
let's consider 4 3 2 1 in base 5. Let's first take a moment and determine each
place value. Remember we can find place values by starting with 5 raising to the
0 power which we see here which gives us the ones. Now if we increase the exponent
by 1 5 to the first would give us the 5s place value. 5 to the second one give us
the 25 s place value. 5 to the third would give us the 125 s place value and
so on. We wanted to write this base 5 number in base 10 we can do so using
each particular place value. This 4 represents 4 5 to the third's or
four 125 s. This 3 represents 3 5 to the seconds or 3 25 s. This 2 represents two 5
to the firsts or just two fives and finally this one represents 1 5 to the
0 or 1 1. So let's simplify on the right side. Notice how we end up with 586 base
10. We have 500 plus 75 plus 10 plus 1. So 4 3 2 1 and base 5 is equal to 586 in
base 10. Let's look at some more examples. We want to convert the base four number
1 0 3 2 1 to base 10. So I think it's always helpful to start with the base
which in this case is 4 and start with 4 to the 0 which we see here work our way
left increasing the exponent on 4 by 1 each time. So we have 4 to the 0, 4 to the
first, four to the second, 4 to the third, and 4 to the fourth. Notice here we have
five place values the same as our number. So to convert this to base 10 we can say
that 1 0 3 2 1 base 4 would be equal to we have this one indicates 1 256. We have
1 times 256 plus we have a 0 in the four to the third or 64 place value let's go
ahead and put that in there we'd have 0 times 4 to the third or 64 plus we have
a 3 in the 4 to the second or 16 place value so we have 3 times 16 plus we have
a 2 in the four the ones place value or just fours place value so we have 2
times 4 plus we have a 1 in the ones place value so we have 1 times 1. And now
we'll simplify to get our base 10 number. We have 256 plus 0 plus 48 plus 8 plus 1
which gives us 313 and this would be base 10 so we found that 1 0 3 2 1 base
4 equals 313 in base 10. Let's take a look at one more example
let's convert the base 7 number 5 6 2 4 to base 10 so we have 7 to the 0, 7 to
the 1, 7 to the 2nd, 7 to the third, and 7 to the 4th. Because our number only has
four digits we only need these 4 place values. So 5 6 2 4 and base 7 is equal to
5 times this would be the 7 to the 3rd place value or they 343 s place value
plus 6 times this would be 7 to the 2nd place value or the 49 s place value plus
2 times 7 to the first or 7 plus 4 times this is the ones place value. And now
we'll simplify 5 times 343 is 1715 plus 6 times 49 is 294 plus 14 plus 4 gives
us 2027. This would be base 10.
So again we can say that five six two four base seven is equal to two thousand
twenty seven in base ten. And we'll go ahead and stop here I hope you found
this helpful.
you