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. - PowerPoint PPT PresentationTRANSCRIPT
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Section 1.1: Integer Operations Section 1.1: Integer Operations and the Division Algorithmand the Division AlgorithmMAT 320 Spring 2008Dr. Hamblin
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AdditionAddition“You have 4 marbles and then you get 7
more. How many marbles do you have now?”
4 711
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SubtractionSubtraction“If you have 9 toys and you give 4 of
them away, how many do you have left?”
9
5
4
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MultiplicationMultiplication“You have 4 packages of muffins, and
each package has 3 muffins. How many total muffins do you have?”
4
3
12
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DivisionDivision“You have 12 cookies, and you want to
distribute them equally to your 4 friends. How many cookies does each friend get?”
12
3
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Examining DivisionExamining DivisionAs you can see, division is the most
complex of the four operationsJust as multiplication is repeated
addition, division can be thought of as repeated subtraction
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28 divided by 428 divided by 428 – 4 = 2424 – 4 = 2020 – 4 = 1616 – 4 = 1212 – 4 = 88 – 4 = 44 – 4 = 0Once we reach 0, we stop. We subtracted
seven 4’s, so 28 divided by 4 is 7.
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92 divided by 1292 divided by 1292 – 12 = 8080 – 12 = 6868 – 12 = 5656 – 12 = 4444 – 12 = 3232 – 12 = 2020 – 12 = 8We don’t have enough to subtract another
12, so we stop and say that 92 divided by 12 is 7, remainder 8.
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Expressing the Answer As an Expressing the Answer As an EquationEquationSince 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.
dividend divisor quotient remainder
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3409 divided by 133409 divided by 13Subtracting 13 one at a time would take a while3409 – 100 · 13 = 21092109 – 100 · 13 = 809809 – 50 · 13 = 159159 – 10 · 13 = 2929 – 13 = 1919 – 13 = 3So 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.
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How Division WorksHow Division WorksStart 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
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Theorem 1.1: The Division Algorithm Theorem 1.1: The Division Algorithm (aka The Remainder Theorem)(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
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Ways to Find the Quotient and Ways to Find the Quotient and RemainderRemainderWe’ve already talked about the repeated
subtraction methodMethod 2: Guess and Check
Fill 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
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Negative NumbersNegative NumbersNotice that in the Division Algorithm, b
must be positive, but a can be negative
How do we handle that?
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-30 divided by 8-30 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
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-30 divided by 8, continued-30 divided by 8, continued-30 + 8 = -22-22 + 8 = -14-14 + 8 = -6 (debt not paid off yet!)-6 + 8 = 2So we made 4 payments and had 2
dollars left over-30 divided by 8 is -4, remainder 2Check: -30 = 8 · (-4) + 2
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Caution!Caution!Negative numbers are tricky, be sure to
always check your answerBe careful when using the calculator
methodExample: -41 divided by 7
The 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