Section 8.1 Modular Arithmetic

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Introduction to Modular Arithmetic

Many clocks have the familiar -hour design. We designate whether the time is before noon or after noon by using the abbreviations A.M. and P.M. A reference to 7:00 A.M. means hours after 12:00 midnight; a reference to 7:00 P.M. means hours after 12:00 noon. In both cases, once is reached on the clock, we begin again with .

Point of Interest

The abbreviation A.M. comes from the Latin ante (before) meridiem (midday). The abbreviation P.M. comes from the Latin post (after) meridiem (midday).

If we want to determine a time in the future or in the past, it is necessary to consider whether we have passed 12 o’clock. To determine the time hours after 3 o’clock, we add and . Because we did not pass 12 o’clock, the time is 11 o’clock (Figure 8.1A). However, to determine the time hours after 9 o’clock, we must take into consideration that once we have passed 12 o’clock, we begin again with . Therefore, hours after 9 o’clock is 5 o’clock, as shown in Figure 8.1B.

Figure 8.1A

Figure 8.1B

We will use the symbol to denote addition on a -hour clock. Using this notation,

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on a -hour clock.

We can also perform subtraction on a -hour clock. If the time now is 10 o’clock, then hours ago the time was 3 o’clock, which is the difference between and . However, if the time now is 3 o’clock, then, using Figure 8.2, we see that hours ago it was 8 o’clock. If we use the symbol to denote subtraction on a -hour clock, we can write

and

Figure 8.2

Example 1

Perform Clock Arithmetic

Evaluate each of the following, where and indicate addition and subtraction, respectively, on a -hour clock.

Solution

Calculate using a -hour clock.

c. https://mindtap.cengage.com/static/nb/ui/evo/index.html?eISBN=9780357048320&id=339189958&snapshotId=876894&dockAppUid=101&nbId=876894& 2/7 3/1/2019 Print Preview d.

Check Your Progress 1

Evaluate each of the following using a -hour clock.

A similar example involves day-of-the-week arithmetic. If we associate each day of the week with a number, as shown below, then days after Friday is Thursday and days after Monday is Wednesday. Symbolically, we write

and

Note: We are using the symbol for days-of-the-week arithmetic to differentiate from the symbol for clock arithmetic.

Another way to determine the day of the week is to note that when the sum is divided by , the number of days in a week, the remainder is , the number associated with Thursday. When is divided by , the remainder is , the number associated with Wednesday. This works because the days of the week repeat every days.

https://mindtap.cengage.com/static/nb/ui/evo/index.html?eISBN=9780357048320&id=339189958&snapshotId=876894&dockAppUid=101&nbId=876894& 3/7 3/1/2019 Print Preview The same method can be applied to -hour-clock arithmetic. From Example 1a, when is divided by , the number of hours on a -hour clock, the remainder is , the time hours after 8 o’clock.

Situations such as these that repeat in cycles are represented mathematically by using modular arithmetic, or arithmetic modulo .

Modulo

Two integers and are said to be congruent modulo , where is a natural number, if is an integer. In this case, we write . The number is called the modulus. The statment is called a congruence.

Example 2

Determine Whether a Congruence Is True

Determine whether the congruence is true.

Solution

a. Find . Because is an integer, is a true

congruence.

b. Find . Because is not an integer, is not a true congruence.

Check Your Progress 2

Determine whether the congruence is true.

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Question

Using the alternate method, is a true congruence?

Now suppose today is Friday. To determine the day of the week days from now, we observe that days from now the day will be Friday, so days from now the day will be Sunday. Note that the remainder when is divided by is , or, using modular notation, . The signiﬁes days after Friday, which is Sunday.

Example 3

A Day of the Week

July 4, 2017, was a Tuesday. What day of the week is July 4, 2022?

Solution

There are years between the two dates. Each year has days except 2020, which has one extra day because it is a leap year. So the total number of days between the two dates is . Because , . Any multiple of days past a given day will be the same day of the week. So the day of the week days after July 4, 2017, will be the same as the day days after July 4, 2017. Thus July 4, 2022, will be a Monday.

Check Your Progress 3

In 2016, Abraham Lincoln’s birthday fell on Friday, February 12. On what day of the week does Lincoln’s birthday fall in 2025?

Math Matters https://mindtap.cengage.com/static/nb/ui/evo/index.html?eISBN=9780357048320&id=339189958&snapshotId=876894&dockAppUid=101&nbId=876894& 5/7 3/1/2019 Print Preview A Leap-Year Formula

The calculation in Example 3 required that we consider whether the intervening years contained a leap year. There is a formula, based on modular arithmetic, that can be used to determine which years are leap years.

The calendar we use today is called the Gregorian calendar. This calendar differs from the Julian calendar (see the Historical Note) in that leap years do not always occur every fourth year. Here is the rule: Let be the year. If , then is a leap year unless . In that case, is not a leap year unless . Then is a leap year.

Using this rule, 2008 is a leap year because , and 2013 is not a leap year because . The year 1900 was not a leap year because but . The year 2000 was a leap year because and .

Historical Note

The congruent symbol consisting of three horizontal bars was introduced in print in 1801 by Carl Friedrich Gauss (1777–1855).

Historical Note

The Julian calendar, introduced by Julius Caesar in 46 BC and named after him, contained months and days. Every fourth year, an additional day was added to the year to compensate for the difference between an ordinary year and a solar year. However, a solar year actually consists of days, not days as assumed by Caesar. The difference between these numbers is days.

Although this is a small number, after a few centuries had passed, the ordinary year and the solar year no longer matched. If something were not done to rectify the situation, after a period of time the summer season in the northern hemisphere would be in December and the winter season would be in July.

To bring the seasons back into phase, Pope Gregory removed days from October in 1582. The new calendar was called the Gregorian calendar and is the one we use today. This calendar did not win general acceptance until 1752.

Chapter 8: Mathematical Systems: Section 8.1 Modular Arithmetic Book Title: Mathematical Excursions Printed By: Jean Nicolas Pestieau ([email protected]) © 2018 Cengage Learning, Cengage Learning https://mindtap.cengage.com/static/nb/ui/evo/index.html?eISBN=9780357048320&id=339189958&snapshotId=876894&dockAppUid=101&nbId=876894& 6/7 3/1/2019 Print Preview

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