Section 1 1 integer operations and the division algorithm
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Section 1.1: Integer Operations and the Division Algorithm. MAT 320 Spring 2008 Dr. Hamblin. Addition. “You have 4 marbles and then you get 7 more. How many marbles do you have now?”. 4. 11. 7. Subtraction. “If you have 9 toys and you give 4 of them away, how many do you have left?”. 5.

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Section 1.1: Integer Operations and the Division Algorithm

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Section 1.1: Integer Operations and the Division Algorithm

MAT 320 Spring 2008

Dr. Hamblin


Addition

  • “You have 4 marbles and then you get 7 more. How many marbles do you have now?”

4

11

7


Subtraction

  • “If you have 9 toys and you give 4 of them away, how many do you have left?”

5

4

9


Multiplication

  • “You have 4 packages of muffins, and each package has 3 muffins. How many total muffins do you have?”

4

12

3


Division

  • “You have 12 cookies, and you want to distribute them equally to your 4 friends. How many cookies does each friend get?”

3

12


Examining Division

  • As you can see, division is the most complex of the four operations

  • Just as multiplication is repeated addition, division can be thought of as repeated subtraction


28 divided by 4

  • 28 – 4 = 24

  • 24 – 4 = 20

  • 20 – 4 = 16

  • 16 – 4 = 12

  • 12 – 4 = 8

  • 8 – 4 = 4

  • 4 – 4 = 0

  • Once we reach 0, we stop. We subtracted seven 4’s, so 28 divided by 4 is 7.


92 divided by 12

  • 92 – 12 = 80

  • 80 – 12 = 68

  • 68 – 12 = 56

  • 56 – 12 = 44

  • 44 – 12 = 32

  • 32 – 12 = 20

  • 20 – 12 = 8

  • We don’t have enough to subtract another 12, so we stop and say that 92 divided by 12 is 7, remainder 8.


Expressing the Answer As an Equation

  • Since 28 divided by 4 “comes out evenly,” we say that 28 is divisible by 4, and we write 28 = 4 · 7.

  • However, 92 divided by 12 did not “come out evenly,” since 92  12 · 7. In fact, 12 · 7 is exactly 8 less than 92, so we can say that 92 = 12 · 7 + 8.

remainder

dividend

quotient

divisor


3409 divided by 13

  • Subtracting 13 one at a time would take a while

  • 3409 – 100 · 13 = 2109

  • 2109 – 100 · 13 = 809

  • 809 – 50 · 13 = 159

  • 159 – 10 · 13 = 29

  • 29 – 13 = 19

  • 19 – 13 = 3

  • So 3409 divided by 13 is 262 remainder 3.

  • All in all, we subtracted 262 13’s, so we could write 3409 – 262 · 13 = 3, or 3409 = 13 · 262 + 3.


How Division Works

  • Start with dividend a and divisor b (“a divided by b”)

  • Repeatedly subtract b from a until the result is less than a (but not less than 0)

  • The number of times you need to subtract b is called the quotient q, and the remaining number is called the remainder r

  • Once this is done, a = bq + r will be true


Theorem 1.1: The Division Algorithm (aka The Remainder Theorem)

  • Let a and b be integers with b > 0. Then there exist unique integers q and r, with 0  r < b and a = bq + r.

  • This just says what we’ve already talked about, in formal language


Ways to Find the Quotient and Remainder

  • We’ve already talked about the repeated subtraction method

  • Method 2: Guess and CheckFill in whatever number you want for q, and solve for r. If r is between 0 and b, you’re done. If r is too big, increase q. If r is negative, decrease q.

  • Method 3: CalculatorType in a/b on your calculator. The number before the decimal point is q. Solve for r in the equation a = bq + r


Negative Numbers

  • Notice that in the Division Algorithm, b must be positive, but a can be negative

  • How do we handle that?


-30 divided by 8

  • “You owe me 30 dollars. How many 8 dollar payments do you need to make to pay off this debt?”

  • Instead of subtracting 8 from -30 (which would just increase our debt), we add 8 repeatedly


-30 divided by 8, continued

  • -30 + 8 = -22

  • -22 + 8 = -14

  • -14 + 8 = -6 (debt not paid off yet!)

  • -6 + 8 = 2

  • So we made 4 payments and had 2 dollars left over

  • -30 divided by 8 is -4, remainder 2

  • Check: -30 = 8 · (-4) + 2


Caution!

  • Negative numbers are tricky, be sure to always check your answer

  • Be careful when using the calculator method

  • Example: -41 divided by 7The calculator gives -5.857…, but if we plug in q = -5, we get r = -6, which is not a valid remainder

  • The correct answer is q = -6, r = 1


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