Question 1

In the last example, the answer is 1. I understand how you got the answer using the clock, but I thought the point of the modulo operator was to find the remainder, and if you divide -5 by 3, the remainder would be 2. Why are these answers different??

Accepted Answer

It is true that 5 divided by 3 gives you a remainder of 2
However, it is NOT true that -5 divided by 3 gives you a remainder of 2. It gives you a remainder of 1.

We can do some long division to prove it:

First we'll do the simple 5/3
```
    _1_R2
3 / 5
   -3   (Since 1 * 3 = 3)
   ---
    2
```
We can check our result:
`5 = 1 * 3 + 2`

Now we can do -5/3
```
    _-2_R1
3 / -5
   -(-6)   (Since -2 * 3= -6)
     --- 
      1
```

We can check our result:
`-5 = -2 * 3 + 1`

The -2 seems strange since we might think 3 goes into 5, -1 times. But this would make our remainder negative (which is not allowed)

Let's see what would happen, if we allowed negative remainders (we don't, but computers do!) it would look like this:
```
   _-1_R-2
3 / -5
   -(-3)     (Since -1 * 3= -3)
   ---
    -2
```

We can check our result:
`-5 = -1 * 3 + (-2)`

Notice that if we go 2 steps counter clockwise on the modular circle for 3 that we end up at 1.
Notice that if we add one multiple of 3 to -2 we end up at 1.
Again, we don't allow negative remainders, but it may give you some intuition for what is going on, and for how congruence modulo works later.

Hope this makes sense

Question 2

Who invented Modular Arithmetic?

Accepted Answer

Johann Carl Friedrich Gauss is usually attributed with the invention/discovery of modular arithmetic. In 1796 he did some work that advanced the field, and in 1801 published the book Disquisitiones Arithmeticae which, amongst other things, introduced congruence modulo and the ≡ symbol. So he is the person that laid out the modern approach to modular arithmetic that we use today.

Gauss is one of the most influential mathematicians of all time. His name is well known amongst mathematicians.

Gauss is one of the most influential mathematicians of all time. His name is well known amongst mathematicians.

Question 3

Is the following method (I devised it by observation ) a known / valid method of getting the negative integers reminder?
if A % B = C
then -A % B = B - C

ex: -19 % 4 can be solved with these steps:
19 % 4 = 3
-19 % 4 = 4 - 3 = 1

Accepted Answer

Yes, it works. It's a cool discovery, but you weren't the first one to discover it.

Here's a simple explanation of how it works.
For our mod n circle.
- positive numbers go clockwise around the circle
- negative numbers go anti-clockwise around the circle
To find where a number has ended up on the circle, in clockwise units,  if you have a number measured in anti-clockwise units you have:
#clockwise_units = size_of_circle - #anti_clockwise_units
Remember that the size of the circle for mod n is n.

Later on, you will find that -x (mod n) is congruent to n-x (mod n).

Hope this makes, and good luck with more discoveries

Question 4

What happens if the modulus is negative?

Accepted Answer

The modulus must be a positive integer i.e. an integer > = 1

Question 5

-17 mod 7

-7*2=-14
-17 '=' -7 *2 +-3

-7 *2 +-3=-17

why does the site say 4? :(

Accepted Answer

You said -7 *2 = -14, but -14 is actually bigger than -17. We have to go a little further. The next step would be -7*3 = -21, which is smaller than -17. Now from -21, how much do we need to add to get to -17?   The answer is 4. Hope this helped! :)

Question 6

how is 3mod10 = 3?? it should be 0..isnt it??

Accepted Answer

3 / 10 = 0 with a remainder of 3
3 mod 10 gives us the remainder when we divide 3 by 10.
Thus 3 mod 10 = 3

Question 7

Can someone explain the concept of calculating mod with out the circles?
How would you do 13 mod 10?

Accepted Answer

13 mod 10.
       1 R 3
10/13
     10
       3
The remainder is 3. 13 mod 10 is 3.
I think this method of modulus is easier than the modulus method. What do you think?

Question 8

There is no video. A congruent  B (modulo C). Does this mean that when A or B is divided by C the remainder will be the same?

Accepted Answer

You are correct when you say
if we have  A ≡ B (mod C) then this will mean  when A or B is divided by C the remainder will be the same

For more info, check out:
https://www.khanacademy.org/math/applied-math/cryptography/modarithmetic/a/congruence-modulo

Question 9

It seems a typo that '(5 is negative)' just under the  -5 mod 3 = ?.  I think 5 is a positive number, -5 is negative. (I have already asked this at Crowdin, but no answer for three weeks. )
Best,
H.

Accepted Answer

Indeed, 5 is a positive number, and -5 is a negative number.

However, for the example:
"-5 mod 3 = ?
With a modulus of 3 we we make a clock with numbers 0,1,2
We start at 0 and go through 5 numbers in counter-clockwise sequence (5 is negative) 2,1,0,2,1"

The "(5 is negative)" is explaining why we are going counter-clockwise as opposed to clockwise 
i.e. if we were looking at 5 mod 3 we would start at 0 and go through 5 numbers in a clockwise sequence, (1,2,0,1,2)  but in the example we have -5 mod 3, so we start at 0 and go through 5 numbers in a counter-clockwise sequence (2,1,0,2,1)

Hope this makes sense

Computer science theory

Course: Computer science theory > Unit 2

What is modular arithmetic?

An Introduction to Modular Math

Visualize modulus with clocks

Examples

$8 mod 4 = ?$ ‍

$7 mod 2 = ?$ ‍

$- 5 mod 3 = ?$ ‍

Conclusion

Notes to the Reader

mod in programming languages and calculators

Congruence Modulo

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