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