I have difficulty understanding Modulo ? What is its meaning?

[kntsy]

I have difficulty understanding "Modulo"? What is its meaning?

$$a\equiv b\left(mod n\right)$$
means a is "congruent" to b "modulo" n
means a-b=knExcept accepting this is an equivalence relation, I feel very uncomfortable about "modulo" because of the lack of visual picture of this concept.
Everytime i have to move the "b" to the left side and see if "k" exists. This is very annoying and it totally shuts down my math intuition.
Why do mathematicians introduce this concept? Will it lead to any profound meaning/result? Also as "a" and "b" can be very far away, why use the word "congruent"?

 

[sjb-2812]

One way to think of modulo arithmetic is like a clock. If it's now 10, the time in 6 hours time need not be 16, but instead 4. Perhaps see http://en.wikipedia.org/wiki/Modular_arithmetic or similar?

 

[HallsofIvy]

$$a\equiv b\left(mod n\right)$$
means a is "congruent" to b "modulo" n
means a-b=kn


Except accepting this is an equivalence relation, I feel very uncomfortable about "modulo" because of the lack of visual picture of this concept.
Everytime i have to move the "b" to the left side and see if "k" exists. This is very annoying and it totally shuts down my math intuition.
Why do mathematicians introduce this concept? Will it lead to any profound meaning/result? Also as "a" and "b" can be very far away, why use the word "congruent"?

"congruent" has nothing to do with be close. "Congruent" means "the same in this particular way" where "this particular way" is defined for that particular use of "congruent".

In "modulo" arithmetic $a\equiv b (mod n)$ if and only if dividing each by n gives the same remainder. $7\equive 19 (mod 4)$ because dividing 7 by 4 gives a quotient of 1 and a remainder of 3 while dividing 19 by 4 gives a quotient of 4 and a remainder of 3. We are ignoring the quotient and looking only at the remainder.

 

[kntsy]

"congruent" has nothing to do with be close. "Congruent" means "the same in this particular way" where "this particular way" is defined for that particular use of "congruent".

In "modulo" arithmetic [itex]a\equiv b (mod n)[/math] if and only if dividing each by n gives the same remainder. [itex]7\equive 19 (mod 4)[/math] because dividng 7 by 4 gives a quotient of 1 and a remainder of 3 while dividing 19 by 4 gives a quotient of 4 and a remainder of 3. We are ignoring the quotient and looking only at the remainder.

thanks hallsofivy. excellent examples and explanations.

 

[CellCoree]

Modulo, or the modulo operation, is a mathematical concept that is used to calculate the remainder after dividing two numbers. It is denoted by the symbol "mod" or "%". For example, 5 mod 2 = 1, as 5 divided by 2 leaves a remainder of 1.

The concept of modulo is used in various areas of mathematics, such as number theory, algebra, and computer science. It has many practical applications, such as in cryptography, coding theory, and data compression.

One of the main reasons mathematicians use the concept of modulo is because it allows for more efficient and concise calculations. It also helps in solving problems involving patterns and cycles. For example, the concept of modulo is used in clock arithmetic, where the hours on a clock repeat in a cycle of 12 or 24.

While the concept of modulo may seem abstract and unfamiliar, it is a fundamental concept in mathematics and has many important applications. With practice, you will become more comfortable with using it and seeing its visual representation. Remember that mathematics is a language and, like any language, it takes time and practice to understand and use it fluently.