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Section 13.3

Math in Our World. Section 13.3. Modular Systems. Learning Objectives. Find congruent numbers in modular systems. Perform addition, subtraction, and multiplication in modular systems. Solve congruences in modular systems. Modular Systems.

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Section 13.3

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  1. Math in Our World Section 13.3 Modular Systems

  2. Learning Objectives • Find congruent numbers in modular systems. • Perform addition, subtraction, and multiplication in modular systems. • Solve congruences in modular systems.

  3. Modular Systems A modular system is a mathematical system with a specific number of elements in which the arithmetic is analogous to the clock arithmetic from Section 13-2. In Section 13-2, we looked very briefly at arithmetic on clocks with numbers of hours other than 12. In this section, these clocks will become our main focus as we define the study of modular arithmetic.

  4. Modular Systems For example, a modular 5 system, denoted as mod 5, has as its five elements 0, 1, 2, 3, 4 and uses the clock shown. A mod 3 system consists of elements 0, 1, 2 and uses the clock shown.

  5. Modular Systems In general, a mod m system consists of m whole numbers starting at zero and ending at m – 1. The number m is called the modulus of the system. The 12-hour clock system we studied in Section 13-2 is a system with modulus 12; the one difference is that we used the whole numbers from 1 through 12 rather than 0 through 11, but the idea is the same.

  6. Congruence a is congruent to b modulo m (written a  b mod m) if a and b have the same remainder when divided by m. Alternately, a  b mod m if m divides b – a. To write a whole number in any given modulo system, we will divide that number by the modulus. The remainder tells us the congruent number in the modular system.

  7. EXAMPLE 1 Finding Congruent Numbers in Modular Systems Find the number congruent to each number in the given modular system. (a) 19 mod 5 (b) 25 mod 3 SOLUTION • Divide 19 by 5 and find the remainder. • So 19  4 (mod 5). This answer can be verified by starting at 0 on the mod 5 clock and counting around 19 numbers.

  8. EXAMPLE 1 Finding Congruent Numbers in Modular Systems SOLUTION (b) Divide 25 by 3 and find the remainder. So 25  1 (mod 3).

  9. Modular Systems To write a negative number in modulo m, you can keep adding m to the negative number until you get a non-negative number. Another way to say this is that you add the smallest multiple of the modulus that yields a nonnegative number.

  10. EXAMPLE 2 Finding a Negative Number in Modulo m Find the number congruent to the given number in the given modular system. • – 13 mod 7 • – 1 mod 3 • – 27 mod 10 • – 20 mod 5

  11. EXAMPLE 2 Finding a Negative Number in Modulo m SOLUTION • Multiples of 7 are 7, 14, 21, etc. • The smallest multiple of 7 that can be added to – 13 that results in a non-negative number is 14. Therefore, – 13 is congruent to – 13 + 14  1 mod 7. • (b) – 1 + 3 = 2, so – 1  2 mod 3 • (c) – 27 + 30 = 3, so – 27  3 mod 10 (since 30 is 3 times the modulus) • (d) – 20 + 20 = 0, so, – 20  0 mod 5 (since 20 is 4 times the modulus)

  12. Arithmetic in Modular Systems The operations of addition, subtraction, and multiplication in modular systems can be performed by adding, subtracting, or multiplying the numbers as usual, then converting the answers to equivalent numbers in the specified system like we did in Examples 1 and 2.

  13. EXAMPLE 3 Adding and Multiplying in Modular Systems Find the result of each operation in the given system • 4 x 4 in mod 5 • 6 + 5 in mod 7 • 5 x 6 in mod 9 • 9 + 7 in mod 8

  14. EXAMPLE 3 Adding and Multiplying in Modular Systems SOLUTION • 4 x 4 = 16 and 16 ÷ 5 = 3 remainder 1: • 4 x 4  1 mod 5. • (b) 6 + 5 = 11 and 11 ÷ 7 = 1 remainder 4: • 6 + 5  4 mod 7. • (c) 5 x 6 = 30 and 30 ÷ 9 = 3 remainder 3: • 5 x 6  3 mod 9. • (d) 9 + 7 = 16 and 16 ÷ 8 = 2 remainder 0: • 9 + 7  0 mod 8.

  15. EXAMPLE 4 Subtraction in Modular Systems Perform the following subtractions in the given modular system. (a) 7 – 10 mod 12 (b) 8 – 3 mod 4 (c) 13 – 21 mod 6 SOLUTION (a) 7 – 10 = – 3; – 3 + 12 = 9, so – 3  9 mod 12 (b) 8 – 3 = 5; so 5  1 mod 4 (c) 13 – 21 = – 8; – 8 + 12 = 4, so – 8  4 mod 6

  16. Solving Congruences An equation like x + 2 = 5 is simple to solve in the real number system because there’s only one number that will result in 5 when added to 2, namely 3. But when we use congruence in modular systems rather than equality in the real number system, the picture is more complicated. In mod 6, for example, the congruence x + 2 = 5 mod 6 has more than one solution. One is x = 3, but x = 9 is also a solution (because 9 + 2 = 5 mod 6), as is x = 15. In fact, we can keep getting new solutions by adding 6 to the previous one.

  17. EXAMPLE 5 Solving a Congruence in a Modular System Find all natural number solutions to 3x – 5  1 mod 6. SOLUTION The first whole number that is congruent to 1 mod 6 is 1, so we begin by solving 3x – 5 = 1.

  18. EXAMPLE 5 Solving a Congruence in a Modular System SOLUTION The next number that is congruent to 1 mod 6 is 7: The next number that is congruent to 1 mod 6 is 13: The pattern emerging is that each new solution is found by adding 2 to the previous solution. So the set of solutions is {2, 4, 6, 8, 10, . . . }.

  19. EXAMPLE 6 Solving a Congruence in a Modular System Find all natural number solutions to 6x  12 mod 3. SOLUTION First, notice that 12 mod 3 is the same as 0 mod 3, so we can rewrite the congruence as 6x  0 mod 3. Next, note that any multiple of 6 will have remainder 0 when dividing by 3; this means that any natural number is a solution, so the solution is {1, 2, 3, 4, 5, . . . }.

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