Section 1.1: Integer Operations and the Division Algorithm MAT 320 Spring 2008 Dr. Hamblin.

Section 1.1: Integer Operations Section 1.1: Integer Operations and the Division Algorithmand the Division AlgorithmMAT 320 Spring 2008Dr. Hamblin

AdditionAddition“You have 4 marbles and then you get 7

more. How many marbles do you have now?”

4 711

SubtractionSubtraction“If you have 9 toys and you give 4 of

them away, how many do you have left?”

9

5

4

MultiplicationMultiplication“You have 4 packages of muffins, and

each package has 3 muffins. How many total muffins do you have?”

4

3

12

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

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

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.

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.

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

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.

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

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

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

Negative NumbersNegative NumbersNotice that in the Division Algorithm, b

must be positive, but a can be negative

How do we handle that?

-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

-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

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