Subtraction in base five
Subtraction in base five is similar to subtraction in base 10. If you understand subtraction in base 10, this lesson will not be hard to understand
Base 10 uses 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9
If you want to perform the following subtraction, what would you do if the numbers are in base 10?
4 8 2

1 2 5
_____________
Since 2 is smaller than 5, you will borrow a ten from 8 tens.
8 tens is now 7 tens. Then, add the ten you borrowed to 2 to get 12
Rewrite the problem as shown below.
7 12
4 8 2

1 2 5
_____________
Base 5 uses 0, 1, 2, 3, and 4
With base 5, you will borrow either a 5, 25, 125, etc depending on the place value not a 10, 100, or 1000 when needed. Why is that?
Some explanations:
In base 10, the number 548 can mean everything you see below:
5 × 10
^{2} + 4 × 10 + 8
5 groups of 100, 4 groups of 10, and 8
5 hundreds + 4 tens + 8
In base 5, the number 243
_{5} could also mean everything you see below:
2 × 5
^{2} + 4 × 5 + 3
2 groups of 25, 4 groups of 5, and 3
2 twentyfives + 4 fives + 3 (I hope an English teacher correct if I am wrong with the grammar)
Simply put, it is because you are in base 5, so any borrowing is done with 5, 25, 125, etc
Say you want to perform the following subtraction in base 5
2 3 1

1 1 4
_____________
Borrow a 5 from 3 fives. 3 fives in now 2 fives
Then, add that 5 to the 1 to make it 6
Rewrite the problem:
2 6
2 3 1

1 1 4
_____________
Now you are ready to do subtraction in base five
Example #1:No carry
To avoid confusion with base 10, we put a 5 next to each number
However, it is clear to you that the subtraction is being done in base five, there is no need to write down the 5
2 3 4_{5}

1 1 3_{5}
_____________________
1 2 1_{5}
The subtraction above is easy since there was no carry
Example #2:Subtraction in base five with carry
4 2 2_{5}

1 4 3_{5}
_____________________
Explanation:
Since you cannot subtract 3 from 2, borrow a 5 from 2 fives. 2 fives is now 1 five
Add 1 five to 2 to make 7
1 7
4 2 2_{5}

1 4 3_{5}
_____________________
Since you cannot subtract 4 from 1, borrow a 25 from 4 twentyfives. 4 twentyfives is now 3 twentyfives
Stop! You cannot just put 25 in the fives place
However, since 25 = 5 × 5, 25 is equal to 5 fives
You can now add 5 fives to 1 five to make it 6 fives
The other way to look at it is to say that right now the 1 in green means that there is a 5
Borrow a 25 and add this to 5 to get 30 and of course 30 is 6 fives
6
3 1 7
4 2 2_{5}

1 4 3_{5}
_____________________
2 2 4_{5}
Now you can safely subtract
7  3 = 4
6  4 = 2
3  1 = 2
Example #3:Subtraction in base five with carry
4 0 0_{5}

2 4 4_{5}
_____________________
You could just borrow a 25 from the twentyfives places.
Since 25 = 5 fives, you could put 5 fives in the fives place
However, there is nothing left to put in the ones place
Therefore borrow instead 2 twentyfives or 10 fives
Then, put 9 fives in the fives place and a five in the ones place
5  4 = 1
9  4 = 5. However, we cannot write 5 in the fives place
Think of it. We never write 10 in the tens place. Instead, we write 0 and carry 1 over to the next higher place.
Since 5 means 5 fives or 1 twentyfive, we can put it in the twentyfive place.
In other words, we write down 0 in the fives place and carry 1 or 1 twentyfive to the twentyfive place.
2 9 5
4 0 0_{5}

2 4 4_{5}
____________________
1 0 1_{5}

Feb 17, 19 12:04 PM
There is no rational number whose square is 2. An easy to follow proof by contraction.
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