Section 1.1: Integer Operations and the Division Algorithm

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

2. Addition • “You have 4 marbles and then you get 7 more. How many marbles do you have now?” 4 11 7

3. Subtraction • “If you have 9 toys and you give 4 of them away, how many do you have left?” 5 4 9

4. Multiplication • “You have 4 packages of muffins, and each package has 3 muffins. How many total muffins do you have?” 4 12 3

5. 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

6. 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

7. 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.

8. 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.

9. 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

10. 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.

11. 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

12. 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

13. 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

14. Negative Numbers • Notice that in the Division Algorithm, b must be positive, but a can be negative • How do we handle that?

15. -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

16. -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

17. 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