How do you do math bases?

Question

Can you please tell me about bases?
For example: 13 base 10= ____ base 5.
Or: 44 (base 5) + 32 (base 5)= ____ (base 5)

Please guide me through these two, tell me how to do it in words, and give another example! Thanks.

Answer 1

Whatever the base is, it's represented by the digits 10. Its square is represented by 100.
So, if the base were 3, we have the numbers
1, 2, and then three itself is represented by 10[3], meaning 1*base + 0 = 1*3 + 0
Four is 11[3] i.e. 1*base + 1
Five is 12[3] i.e. 1*base + 2
Six is 20[3] i.e. 2*base + 0 = 2*3 + 0
and so on: 21, 22, then 100 meaning
1*(base)^2 + 0*base + 0
= 1*9 +0 + 0

In base 3, 212 means
2*(base)^2 + 1*base + 2
= 2*9 + 1*3 + 2
which is eighteen + three + two = twenty three.

To change to base 5, divide the number by 5:
13/5 gives 2 with remainder 3.
That means 13 = 2*5 + 3 = 2*base + 3, so you write it as
23 [base 5]

For the addition, first add the units digits: 4+2 = 6, which is
1*5 + 1 and so that's 11 [base 5].
Put 1 in the units place and carry the other 1 into the second column (the fives)
Then add that 1 to 4+3, getting 8,
which is 1*5 + 3, therefore 13 [base 5]
Write that 3 in the "fives" column, and carry the 1 over into the third (the "twenty fives") column.
So the answer is 131 [base 5]

To check, we know
44 [base 5] = 4*5 + 4 = 24,
and 32 [base 5] = 3*5 + 2 = 17

24 + 17 = 41
and 131 [base 5] = 1*25 + 3*5 + 1
= 41, so it's correct.

You want me to do another example, or give you one to try?
Try this:

17 [base ten] = _______ [base 5]
9 [base ten] = ________ [base 5]
38 [base ten] = _______ [base 5]
61 [base ten] = _______ [base 5]

44 [base 5] + 23 [base 5] = ______ [base 5]

The answers are below, if you want to check, but try not to look before doing some yourself. Want to know why I write ten instead of 10? Because the whole point is that 10 doesn't always mean ten -- only in base ten. In base 8, 10 means eight, in base 5, 10 means five, etc. But I'm being a bit pedantic, and I don't know anyone else who bothers to do it that way.








32 [base 5], 14 [base 5], 123 [base 5], 221 [base 5]

122 [base 5]

Answer 2

I'll start with base 5. Base 5 counts like this: 1,2,3,4,10,11,12,13,14,20. Basically 10 in Base 5 means 5, 11 in base 5 means 6, 20 would mean 10, 30 would mean 15, etc. Each digit place can only count up to the base number-1 before needing to shift to the next decimal place. The examples should show what I mean.

so 13 in base 10 means 1 10 + 3 1's. To get to 13 in base 5 you need 2 5's plus 3 1's. Thus the answer to the first question is 23.

For the second question, these are both in base 5 so they can be directly added, but remember that every digit place can only count up to 4. So 44 base 5 + 32 base 5 = 76 base five. Since you can only count up to 4 in each place, this is an improper answer. In base 5, every 1 means 5 in the previous decimal place, so the proper answer is 131. This means (1 (5x5)'s + 3 5's + 1 1's or 25+15+1=41 in base 10. To check this, 44 base 5 = 4 5's +4 or 24. 32 base 5 means 3 5's plus two ones, or 17. 17+24 = 41, so the answer checks.

Another example would be 76 base 8 = ? in base 6. I always find it easiest to go to base 10 first and then back, so...

76 base 8 = 7 8's + 6 1's or 57. To get to 57 in base 6, I need 1 36 (6x6) + 3 6;s and 3 1's, so this would be 133.

Perhaps someone else can explain this more clearly, but I hope this helps!

Answer 3

13 base 10= ____ base 5
In base ten, you have hundreds, tens and ones (325 is 3 hundreds, 2 tens, and 5 ones)
In base 5, you have twenty-fives, fives and ones.

To change 13 from base 10 to base 5, divide by 25 (won't go), so zero in the twenty-fives place.
Now divide by 5, 13/5 = 2 and a remainder, so 2 goes in the fives place. 13 - (5*2) = 13-10 = 3
So 3 goes in the ones place.
13 base10 = 23 base5

Let's do a conversion from another base into base10.
125 Base8 = ? Base10
1 is in the 64s place (8squared), 2 is in the 8s place (8^1) and 5 is in the ones place.
1*8^2 + 2*8^1 + 5 = 64 + 2*16 + 5 = 101 base10

44 (base 5) + 32 (base 5)= ____ (base 5)
Adding is the same as in base10, except in base10 you carry when you have a sum greater than 10 and in base5 you carry when you have a sum greater than 5.

44
+ 32
------
76 base5
but 6 is 11 in base5, so change 76 to 81 (carry one over to the seven)
Also 8 is 13 in base5, so change 81 to 131 (carry one over to the 25s place).
The answer is 131 base5.

Let's check.
44 base5 = (4*5 +4) base10 = 24 base10
32 base5 = (3*5 + 2) base10 = 17 base10
(24+17) base10 = 41 base10
Now convert to base5.
First divide 41 by 25 (that's 5^2). It goes one time, with a remainder.
(41-25 = 16). Divide 16 by 5. It goes 3 times with a remainder of one.
So the answer is 131 base5

Let's try something in base3.
What's (202 - 111) base3 ?
in the ones place, you have 2-1 = 1, so no problem
in the threes place, you have 0-1, so you need to borrow three from the nines place.
that changes the problem to 132-111 base3 = 21 base3

Check in base10.
202 base3 = 2*3^2 + 0*3^1 + 2 = 18+2 = 20 base10
111 base3 = 1*3^2 + 1*3^1 + 1 = 9+3+1 = 13 base10
21 base3 = 2*3^1 +1 = 6+1 = 7 base10
20 - 13 = 7 check

Answer 4

Bases are quite easy. You use base 10 all the time probably without knowing it. In a base N system the rightmost digit represents the number of N^0 terms, the second to the right is the number of N^1 terms, the third to the right is the number of N^2 terms, etc.... Any single digit may be at most (N-1)... for example, the number 146 in base 10 means 1 10^2 + 4 10^1 + 6 10^0. that is, 100 + 40 + 6. Additionally, since it's base 10, then each digit can take on the values 0-9 only....

So 13 base 10 is just the 'normal' meaning, so you need only accumulate 13 in terms of powers of five. Start with the largest power of five you can fit into the number you're converting.... since 5^2 is 25, you cannot use that term.... next is 5^1 which is less than 13... you can use that term.... how many 5^1 terms fit into 13? two... so your first digit is 2. Now what's left is 3. Since 5^1 is now too large for that, move to the 5^0 term (5^0 =1)... leaving you with 3 for the next digit. Thus the answer is 23 base 5 (usually notated with a subscripted 5 after the 23 to represent the base)

Now the next one is just adding in base five. This works just as in base 10, except your carry points will be a little different. Working from right to left just as in base 10, 2+4 = 11 base 5 (see how that works? it would be six, but six is too large to represent in base 5, so you make 1 5^1 + 1 5^0). Carry the 1, so the next digit is 1 +4+3 base 5 which is 13. Then just as in base 10 addition the 1 falls down... giving a final answer of 131.

If you like computers at all, you can also apply this to binary which is just a fancy word for base 2 arithmetic. so you get things like 110 + 111 = 1101.

Answer 5

Base ten is basically counting with decimals. You have three In the ones place and one in the tens place. Therefore, you have 3 1's and 1 10, basic kindergarton stuff. This also equals 3*1+1*10=13. If you have 23 base five then you would instead move up by 5's, so you have 3 in the ones place, and 2 in the 5's place. So you have 3*1+2*5, which equals 13! so 13 base 10= 23 base five.

I do not know the trick for adding with bases, so I just solve as follows:

(44 base 5 = 4*1+4*5=24)+(32 base 5 = 2*1+3*5=17)=_____ base 5
24+17= _____ base 5
41=_____ base 5
1*1+x*5=41
1+5x=41
5x=40
x=8
1*1+8*5=41
81 base 5 = 41

If you had 215 base 10= _____ base 5:
5*1+x*5+y*25=215
5x+25y=210
5x= 10
x=2
25y=200
y=8

215 base 10= 825 base 5

Answer 6

In base ten, the first digit is how many groups of 1's, the second digit is how many groups of 10, the third is how many groups of 100 (10x10), the fourth is how many groups of 1000(10x10x10), etc. So in base 10, the number 7356 is equal to "7 groups of 1000, 3 groups of 100, 5 groups of 10, and 6 groups of 1" yes?

In base five, the first digit is groups of 1, the second digit is groups of 5, the third digit is groups of 25 (5x5), the fourth digit is groups of 125 (5x5x5), etc. So in base 5, the number 4342 is equal to "4 groups of 125, 3 groups of 25, 4 groups of 5, and 2 groups of 1." This would be 500 + 75 + 20 + 2, or 597.

That works with any base. In base 3, you have groups of 1, groups of 3, groups of 9, groups of 27, groups of 81, etc. In base 2, you have groups of 1, groups of 2, groups of 4, groups of 8, etc.

So we have 13 (in base 10). What is that in base 5? Well, 13 is two groups of 5 and 3 groups of 1, yes? That means that it is 23 in base 5. If you had 56 in base 10 and you wanted to convert it to base 5, you would say "we have 2 groups of 25, which is 50, and six left over. Six is 1 group of 5, with one left over." So the final answer is 211. Two groups of 25, one group of 5, and 1 left over.

To add 44(base5) and 32(base5) you can do one of two things. First, convert them to base 10, add them, and then convert the answer back to base 5. Let's do that: 44 is 4 groups of 5 and 4 left over, which is 24(base10). 32 is 3 groups of 5 and 2 left over, which is 17(base10). 24 and 17 is 41. Covert that back to base 5: 41 is 8 groups of 5 and 1 left over, so 41(base10) is 81(base5).

The other way is to write 44 and 32 over top of each other, just like you're adding normally. Then add the 1's together, and you get 6. But 6 is really 11 in base five (one group of five and one left over), so you put the 1 down and carry the 1. Now add the next column, the 25's column: 4 plus 3 is seven, and 1 more makes 8. Answer: 81.

It would be much easier to show you with pencil and paper, since I write while I talk. But, I hope this helps!