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You may see modulo operations on numbers expressed as either of the following
- a modulo n
- a mod n (abbreviated version)
Example Math Problems
17 modulo 3
- 17 - 3 = 14
- 14 - 3 = 11
- 11 - 3 = 8
- 8 - 3 = 5
- 5 - 3 = 2
20 mod 5
- 20 - 5 = 15
- 15 - 5 = 10
- 10 - 5 = 5
- 5 - 5 = 0
Modulo: Definition, How it Works, and Real-Life Uses
- Guide Authored by Corin B. Arenas, published on October 24, 2019
Most people haven't heard of modular arithmetic or mod outside of math class.
However, if you've ever estimated lunch for 10 people, and found that there's a lot of food leftover, you're actually dealing with a mod problem. People use modular arithmetic all the time, especially with anything involving remainders, time and calendar schedules.
In this section, you'll learn about modulo, its basic operation, and its uses in real life.
What is Modulo?
Modular arithmetic, sometimes called clock arithmetic, is acalculation that involves a number that resets itself to zero each time a wholenumber greater than 1, which is the mod, is reached. An example of this is the24-hour digital clock, which resets itself to 0 at midnight.
In mathematics, the modulo is the remainder or the number that's left after a number is divided by another value. Modulo is also referred to as ‘mod.'
The standard format for mod is:
a mod n
Where a is the value that is divided by n.
For example, you're calculating 15 mod 4. When you divide 15 by 4, there's a remainder.
15 / 4 = 3.75
Instead of its decimal form (0.75), when you use the mod function in a calculator, the remainder is a whole number. For this example, 15 / 4 = remainder 3, which is also 15 = (4 * 3) + 3. Here's how to calculate it manually:
15 mod 4
15 – 4 = 11
11 – 4 = 7
7 – 4 = 3
Calculating Mod with a Negative Number
One might presume the mod function generates the same values as positive numbers when one number is negative. This is actually not the case.
For instance, if you have 340 mod 60, the remainder is 40.
But if you have -340 mod 60, the remainder is 20.
Why does this happen? Mathforum.org explains, with a positive number like 340, the multiple subtracted is less than the absolute value, which results in 40.
340 mod 60
340 – 60 = 280
280 – 60 = 220
220 – 60 = 160
160 – 60 = 100
100 – 60 = 40
But with -340, we subtract a number with a greater absolute value, so the mod function generates a positive value. The resulting remainder is also smaller compared to when both numbers are positive.
Here's how to solve mod with a negative number:
a mod n is a/n = r (remainder)
Therefore, a mod n = a – r * n
Take note: When we input a/b in a calculator, we take the decimal part of the generated value, and round it up to the next integer. Let's do it with the example below:
-340 mod 60
-340/60 = 5.6, when we take the decimal part, it becomes the integer -6
= -340 -(-6) * 60
= -340 -(-360)
= 20
To help you visualize, the number line below shows the difference in value.
Who Created Modular Arithmetic?
According to Britannica, the concept of modular arithmetic has been used by ancient civilizations such as the Indians and Chinese. An example is the Chinese book Master Sun's Mathematical Manual, which dates back from 300 AD.
Moreover, modular arithmetic was used to solve astronomical and seasonal calculations which were problems associated with natural and man-made cycles.
Carl Friedrich Gauss and the Number Theory
In Western mathematics, German mathematician and physicist Carl Friedrich Gaussdid the first systematic study of modular arithmetic. Gauss is regarded as oneof the most influential figures in modern mathematics.
In his early 20s in 1801, he published Disquisitiones Arithmeticae, which laid the foundation for today's number theory and showed the first proof of the law of quadratic reciprocity.
In the number theory, scholars analyze the properties of naturalnumbers, which are whole numbers like -1, -2, 0, 1, 2, and so on. Theirobjective is to discover unexpected mathematical patterns and interactionsbetween natural numbers.
Britannica notes that in modular arithmetic, where mod is N,all the numbers (0, 1, 2, …, N − 1,) are known as residues modulo N. Theresidues are added by finding the arithmetic sum of the numbers, and the mod issubtracted from the sum as many times as possible. This diminishes the sum to anumber M, which is between 0 and N – 1.
In his book, Gauss included a notation with the symbol ≡,which is read as 'is congruent to.' Instead of the usual = symbol, the threehorizontal line segments both signify equality and definition.
For instance, if we add the sum of 2, 4, 3 and 7, the sum is congruent to 6 (mod 10). That's 16 ≡ (mod 10). This means 16 divided by 10 leaves a remainder of 6. Likewise, 16 – 10 = 6.
Another example, 13 ≡ 1 (mod 12). This means 13 divided by 12 leaves a remainder of 1. Likewise, 13 – 12 = 1.
What are Real-World Uses for Mod?
For practical applications, mod is especially useful for dealingwith time.
Since we have 24 hours in a day, it makes sense to refer totime in a 24-hour fashion. This is the principle behind the military time system, beginningat midnight with 0000 hours, and ending the hour at 11PM with 2300 hours.
Instead of saying 9 o'clock PM, they say 2100 hours. Themilitary uses this to coordinate with bases and other personnel located indifferent time zones. Moreover, all pilots (commercial or otherwise) use the 24-hourclock to avoid confusion while traveling between time zones.
To set a standard, pilots and the military use the Greenwich Mean Time (GMT) which they also call Zulu time (Z). For instance, when pilots report that a plane will reach a base at 2100Z, it means it will arrive at 9PM GMT.
Needs Mod Sims 4
How is this connected with modulo? For people stayingin one time zone, it's more important to tell time by separating night and day.This is why the 12-hour standard time uses modulo.
Sim 4 Mod Pack Free
Instead of saying 1600 hours, we just say 4 o'clock. The12-hour standard time uses mod 12 so that 1600 hours becomes 4 o'clock.
The Sims 4 Mod That Work
When we make appointments, it's generally understood peoplemean 4 in the afternoon. Unless specified otherwise, a 4am meeting is absurd,unless you work at night and have online meetings with clients from other time zones.
Organizing Books, Bank Info, and Housing Loan Rates
Mod is useful for organizing large information. Books aretracked using modular arithmetic to calculate checksums for international standardbook numbers (ISBN). In 2007, a 13-digit ISBN number system(which was previously 10) was introduced to help manufacturers identify a largevolume of books.
The same principle is also used by banks to identify errors oninternational bank account numbers (IBAN) when they track transactions fromother countries.
When it comes to housing loans, mod is used to reset calculations for a new period. For instance, a 5/6 adjustable rate mortgage (ARM) resets its interest rates periodically every 6 months. Mod is used to adjust the rates accordingly.
Cryptography and Computer-Generated Art
Modular arithmetic has other applications in the field of cryptography, art and graphics design.
For many years, artists have been using mathematical shapes based on formulas to create designs. Today, the same concept is applied to computer graphics, as well as sculptures and modern paintings.
In cryptography,codes are written to protect secret data. Cryptographers use mod in the Diffie-HellmanKey Exchange in setting up SSL connections to encrypt web traffic.
Encryption is important because it allows users to safeguard information. That's why your personal emails, credit card number, and other personal details should be encrypted whenever you send information on the internet.
The Bottom Line
Mod is a mathematical function that allows us to measure the remainder in a sum. We use this fundamental concept whenever we tell time.
The concept of modular arithmetic has been used by ancientChinese and Indians for centuries. But it was introduced into Westernmathematics by German scientist Carl Friedrich Gauss, who also developed thebasis for the number theory.
Real-world uses for mod include organizing ISBN and bank information, resetting ARM rates, computer graphics design, and cryptography which helps protect private data.
About the Author
Corin is an ardent researcher and writer of financial topics—studying economic trends, how they affect populations, as well as how to help consumers make wiser financial decisions. Her other feature articles can be read on Inquirer.net and Manileno.com. She holds a Master's degree in Creative Writing from the University of the Philippines, one of the top academic institutions in the world, and a Bachelor's in Communication Arts from Miriam College.
Warning: We are not responsible for any issue which may affect in-game issue, game saves, custom contents or game files. Best program for painting. Read and follow the instructions on the links to ensure stable gaming experience. Do this at your own risk.
How To Remove Mosaic or Censorship
The No Mosaic/Censor Mod is perfect
In mathematics, the modulo is the remainder or the number that's left after a number is divided by another value. Modulo is also referred to as ‘mod.'
The standard format for mod is:
a mod n
Where a is the value that is divided by n.
For example, you're calculating 15 mod 4. When you divide 15 by 4, there's a remainder.
15 / 4 = 3.75
Instead of its decimal form (0.75), when you use the mod function in a calculator, the remainder is a whole number. For this example, 15 / 4 = remainder 3, which is also 15 = (4 * 3) + 3. Here's how to calculate it manually:
15 mod 4
15 – 4 = 11
11 – 4 = 7
7 – 4 = 3
Calculating Mod with a Negative Number
One might presume the mod function generates the same values as positive numbers when one number is negative. This is actually not the case.
For instance, if you have 340 mod 60, the remainder is 40.
But if you have -340 mod 60, the remainder is 20.
Why does this happen? Mathforum.org explains, with a positive number like 340, the multiple subtracted is less than the absolute value, which results in 40.
340 mod 60
340 – 60 = 280
280 – 60 = 220
220 – 60 = 160
160 – 60 = 100
100 – 60 = 40
But with -340, we subtract a number with a greater absolute value, so the mod function generates a positive value. The resulting remainder is also smaller compared to when both numbers are positive.
Here's how to solve mod with a negative number:
a mod n is a/n = r (remainder)
Therefore, a mod n = a – r * n
Take note: When we input a/b in a calculator, we take the decimal part of the generated value, and round it up to the next integer. Let's do it with the example below:
-340 mod 60
-340/60 = 5.6, when we take the decimal part, it becomes the integer -6
= -340 -(-6) * 60
= -340 -(-360)
= 20
To help you visualize, the number line below shows the difference in value.
Who Created Modular Arithmetic?
According to Britannica, the concept of modular arithmetic has been used by ancient civilizations such as the Indians and Chinese. An example is the Chinese book Master Sun's Mathematical Manual, which dates back from 300 AD.
Moreover, modular arithmetic was used to solve astronomical and seasonal calculations which were problems associated with natural and man-made cycles.
Carl Friedrich Gauss and the Number Theory
In Western mathematics, German mathematician and physicist Carl Friedrich Gaussdid the first systematic study of modular arithmetic. Gauss is regarded as oneof the most influential figures in modern mathematics.
In his early 20s in 1801, he published Disquisitiones Arithmeticae, which laid the foundation for today's number theory and showed the first proof of the law of quadratic reciprocity.
In the number theory, scholars analyze the properties of naturalnumbers, which are whole numbers like -1, -2, 0, 1, 2, and so on. Theirobjective is to discover unexpected mathematical patterns and interactionsbetween natural numbers.
Britannica notes that in modular arithmetic, where mod is N,all the numbers (0, 1, 2, …, N − 1,) are known as residues modulo N. Theresidues are added by finding the arithmetic sum of the numbers, and the mod issubtracted from the sum as many times as possible. This diminishes the sum to anumber M, which is between 0 and N – 1.
In his book, Gauss included a notation with the symbol ≡,which is read as 'is congruent to.' Instead of the usual = symbol, the threehorizontal line segments both signify equality and definition.
For instance, if we add the sum of 2, 4, 3 and 7, the sum is congruent to 6 (mod 10). That's 16 ≡ (mod 10). This means 16 divided by 10 leaves a remainder of 6. Likewise, 16 – 10 = 6.
Another example, 13 ≡ 1 (mod 12). This means 13 divided by 12 leaves a remainder of 1. Likewise, 13 – 12 = 1.
What are Real-World Uses for Mod?
For practical applications, mod is especially useful for dealingwith time.
Since we have 24 hours in a day, it makes sense to refer totime in a 24-hour fashion. This is the principle behind the military time system, beginningat midnight with 0000 hours, and ending the hour at 11PM with 2300 hours.
Instead of saying 9 o'clock PM, they say 2100 hours. Themilitary uses this to coordinate with bases and other personnel located indifferent time zones. Moreover, all pilots (commercial or otherwise) use the 24-hourclock to avoid confusion while traveling between time zones.
To set a standard, pilots and the military use the Greenwich Mean Time (GMT) which they also call Zulu time (Z). For instance, when pilots report that a plane will reach a base at 2100Z, it means it will arrive at 9PM GMT.
Needs Mod Sims 4
How is this connected with modulo? For people stayingin one time zone, it's more important to tell time by separating night and day.This is why the 12-hour standard time uses modulo.
Sim 4 Mod Pack Free
Instead of saying 1600 hours, we just say 4 o'clock. The12-hour standard time uses mod 12 so that 1600 hours becomes 4 o'clock.
The Sims 4 Mod That Work
When we make appointments, it's generally understood peoplemean 4 in the afternoon. Unless specified otherwise, a 4am meeting is absurd,unless you work at night and have online meetings with clients from other time zones.
Organizing Books, Bank Info, and Housing Loan Rates
Mod is useful for organizing large information. Books aretracked using modular arithmetic to calculate checksums for international standardbook numbers (ISBN). In 2007, a 13-digit ISBN number system(which was previously 10) was introduced to help manufacturers identify a largevolume of books.
The same principle is also used by banks to identify errors oninternational bank account numbers (IBAN) when they track transactions fromother countries.
When it comes to housing loans, mod is used to reset calculations for a new period. For instance, a 5/6 adjustable rate mortgage (ARM) resets its interest rates periodically every 6 months. Mod is used to adjust the rates accordingly.
Cryptography and Computer-Generated Art
Modular arithmetic has other applications in the field of cryptography, art and graphics design.
For many years, artists have been using mathematical shapes based on formulas to create designs. Today, the same concept is applied to computer graphics, as well as sculptures and modern paintings.
In cryptography,codes are written to protect secret data. Cryptographers use mod in the Diffie-HellmanKey Exchange in setting up SSL connections to encrypt web traffic.
Encryption is important because it allows users to safeguard information. That's why your personal emails, credit card number, and other personal details should be encrypted whenever you send information on the internet.
The Bottom Line
Mod is a mathematical function that allows us to measure the remainder in a sum. We use this fundamental concept whenever we tell time.
The concept of modular arithmetic has been used by ancientChinese and Indians for centuries. But it was introduced into Westernmathematics by German scientist Carl Friedrich Gauss, who also developed thebasis for the number theory.
Real-world uses for mod include organizing ISBN and bank information, resetting ARM rates, computer graphics design, and cryptography which helps protect private data.
About the Author
Corin is an ardent researcher and writer of financial topics—studying economic trends, how they affect populations, as well as how to help consumers make wiser financial decisions. Her other feature articles can be read on Inquirer.net and Manileno.com. She holds a Master's degree in Creative Writing from the University of the Philippines, one of the top academic institutions in the world, and a Bachelor's in Communication Arts from Miriam College.
Warning: We are not responsible for any issue which may affect in-game issue, game saves, custom contents or game files. Best program for painting. Read and follow the instructions on the links to ensure stable gaming experience. Do this at your own risk.
How To Remove Mosaic or Censorship
The No Mosaic/Censor Mod is perfect
for those who prefer to see their Sims without censors when doing certain actions. On Mod the Sims, Moxiemason posted the No Mosaic/Censor Mod for The Sims 4 from a Web site which allows removal of the mosaic effect when Sims perform certain actions such as peeing or taking a shower.
The Sims 4 24 Hour Clock Modern
The mod simply edits the resource '0x545AC67A 0x00D4FA33 0xDEAD4915C9C6C0D8' file located in the ClientFullBuild0.package. As far as the uploading goes, it should not have any conflict with the game or other mods since it is an independent file.
Players can download the file then extract it to the Electronic Arts/The Sims 4/Mods folder on their computer. The uploader encourages players to report any issue found using this mod.
Alarm Clock Mod Sims 4
Time Speed Changer
Another mod available for 'The Sims 4' players is the 'Time Speed Changer - Make the Game Clock Run Slower' uploaded by IceM, also on Mod The Sims. Through it, you get to play according to the real-world time instead of the default in-game speed.
According to the uploader, the mod is designed for those who want to play the game realistically in terms of time, controlling smaller families, with free will turned off among Sims. Here are some examples and/or benefits players can enjoy when the mod is applied:
- Having enough time to prepare before going on a date with another Sim
- Being able to prepare everything before throwing a party for your Sim
- Having enough time to finish important tasks
If applied, the time is set to 200 milliseconds per Sim second, equivalent to 12 seconds per Sim minute. Players can also modify the time variable by changing the '.py' file inside the zip file which can be downloaded here.
