- #1
chwala
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- Homework Statement
- how is -10≡2 mod 3?
- Relevant Equations
- modular arithmetic
ok this is a bit confusing to me, long since i did this things...-10/3=quotient -3+ remainder -1. How is the remainder 2?
i am not using any , is it that a=b mod c and a≡ b mod c imply different things? my interest is on the former. The latter i presume applies to congruence.Math_QED said:By definition, this means that ##3## divides ##-10-2 = -12## which is trivially true.
What definition are you using?
chwala said:Problem Statement: how is -10≡2 mod 3?
Relevant Equations: modular arithmetic
ok this is a bit confusing to me, long since i did this things...-10/3=quotient -3+ remainder -1. How is the remainder 2?
If you want to talk about remainders in connection to modular arithmetic you should define the remainder of n/m to be the smallest non-negative number k such that the quotient (n-k)/m is an integer. With this definition you have the quotient -4 and the remainder 2.chwala said:Problem Statement: how is -10≡2 mod 3?
Relevant Equations: modular arithmetic
ok this is a bit confusing to me, long since i did this things...-10/3=quotient -3+ remainder -1. How is the remainder 2?
The reason is that multiples of three doesn't count anymore. The procedure is, that we impose a relation ##\sim## on the set of integers which is defined by: ##a\sim b## iff ##3## divides ##a-b## or iff ##a## and ##b## have the same sort of remainder: ##3\mathbb{Z}##, ##3\mathbb{Z}+1##, or ##3\mathbb{Z}+2##. We consider these sets as one element: ##[0]=3\mathbb{Z}##, ##[1]=3\mathbb{Z}+1##, ##[2]=3\mathbb{Z}+2##. The brackets ##[]## which indicates that entire sets are behind the numbers ##0,1,2## are usually not written, as they don't carry any additional information compared to what has already been said by using ##\operatorname{mod}## or ##\equiv##. The same is true for ##\equiv##: it is often just written as equality ##=##, an equality of sets, the so called equivalence classes with respect to ##\sim##.chwala said:Problem Statement: how is -10≡2 mod 3?
Relevant Equations: modular arithmetic
ok this is a bit confusing to me, long since i did this things...-10/3=quotient -3+ remainder -1. How is the remainder 2?
In addition to what has already been stated (or perhaps I missed that this, too, has already been stated):chwala said:Problem Statement: how is -10≡2 mod 3?
Relevant Equations: modular arithmetic
ok this is a bit confusing to me, long since i did this things...-10/3=quotient -3+ remainder -1. How is the remainder 2?
Orodruin said:If you want to talk about remainders in connection to modular arithmetic you should define the remainder of n/m to be the smallest non-negative number k such that the quotient (n-k)/m is an integer. With this definition you have the quotient -4 and the remainder 2.
Also, as has been pointed out already, -1 = 2 mod 3 so saying -10 = -1 mod 3 is also perfectly fine, -1 and 2 are in the same equivalence class.
so you're implying that k cannot be a negative integer, rather it has to be positive?Orodruin said:If you want to talk about remainders in connection to modular arithmetic you should define the remainder of n/m to be the smallest non-negative number k such that the quotient (n-k)/m is an integer. With this definition you have the quotient -4 and the remainder 2.
Also, as has been pointed out already, -1 = 2 mod 3 so saying -10 = -1 mod 3 is also perfectly fine, -1 and 2 are in the same equivalence class.
i don't understand what you're saying here on inverses...this is a relative new area to me as my area of maths is applied maths...PeroK said:##-1## means the inverse of ##1##. In mod 3 arithmetic ##1 + 2 = 0##, hence ##-1 = 2##.
More simply, ##-1 \equiv 2 (mod \ 3)##
chwala said:i don't understand what you're saying here on inverses...this is a relative new area to me as my area of maths is applied maths...
No, both -10 = -1 mod 3 and -10 = 2 mod 3 are valid statements as -1 = 2 mod 3.chwala said:so you're implying that k cannot be a negative integer, rather it has to be positive?
This is just the definition of a ~ k (mod b). See, for example, post #7.chwala said:i figured out a way of doing this things...lets say you have a≡ k mod b then it follows that,
{a-k}/{b}= integer value...positive critism is welcome.
thanks seen, i appreciate.Orodruin said:This is just the definition of a ~ k (mod b). See, for example, post #7.
Yes, this is correct.chwala said:i figured out a way of doing this things...lets say you have a≡ k mod b then it follows that,
{a-k}/{b}= integer value...positive criticism is welcome.
This is wrong. The representatives (remainders) do not have to be positive. This is out of convenience and not a mathematical necessity. Thus it doesn't follow from anything. One can perfectly calculate with ##\mathbb{Z}/2\mathbb{Z} =\{\,[17],[-12]\,\}## but it makes more fun to write it as ##\{\,[0],[1]\,\}##.MidgetDwarf said:The reason that the reduced residue (the remainder) must be positive follows from the statement of the Division Theorem.
fresh_42 said:This is wrong. The representatives (remainders) do not have to be positive. This is out of convenience and not a mathematical necessity. Thus it doesn't follow from anything. One can perfectly calculate with ##\mathbb{Z}/2\mathbb{Z} =\{\,[17],[-12]\,\}## but it makes more fun to write it as ##\{\,[0],[1]\,\}##.
Right. I just wanted to prevent the impression that it has to be those representatives. This contradicted the entire concept of equivalence classes. Any are as good as specific ones, just not as easy to calculate with. And if we only consider the group property and forget about the ring, then ##\mathbb{Z}_2=\{\,0,1\,\}## can even be written multiplicatively as ##\mathbb{Z}_2=\{\,-1,1\,\}##.MidgetDwarf said:writing the remainders as positive helps us build more results of number theory in a "nice way."
yapPeroK said:Do you understand why:
##-1 \equiv 2 (mod \ 3)##?
Be careful! You will never again agree with people who say ##1+1=2## and expect this as a given truth!chwala said:i took my time to look at the modular arithmetic and i could understand it straight away...i found it to be very easy stuff for me...i believe i can attempt any problem in math if i give it the necessary time and effort.
Modular arithmetic is a branch of mathematics that deals with operations on integers and their remainders when divided by a fixed number, called the modulus.
To calculate the remainder 2 in modular arithmetic, we need to divide the given integer by 2 and take the remainder. For example, the remainder 2 of 13 in modular arithmetic would be 1, as 13 divided by 2 leaves a remainder of 1.
The remainder 2 in modular arithmetic is significant because it indicates whether a number is even or odd. If a number has a remainder of 2 when divided by 2, it is an odd number. If it has a remainder of 0, it is an even number.
Yes, modular arithmetic can be applied to negative numbers. In this case, the remainder is always positive and less than the modulus. For example, the remainder of -5 when divided by 2 would be 1, as -5 divided by 2 leaves a remainder of -1, which is then converted to a positive remainder of 1.
Modular arithmetic has numerous real-world applications, such as in cryptography, computer science, and engineering. It is used to ensure data security, create unique identification numbers, and optimize computer algorithms, among other things.