1. Write the Roman numeral for the Hindu-Arabic numerals: 5, 11, 49, 1328, 1900, 1998 and the year you were born.
2. Write the Hindu-Arabic numeral for the Roman numerals: IV, XIV, LVII, XLIX, MDCCCXLVIII and MCMXLI.
3. What is the value of the place where 1 resides in the numerals: 321, 418, 1234, 7182603, 123456708, 3214, 4189, 12345, 51426037, 10002?
4. Without using more than nine $1 bills, nine $10 bills, nine $100 bills and nine $1,000 bills, how would you make: $427, $5612?
5. Without using more than four $1 bills, four $5 bills and four $20 bills, how would you make $17, $24, $101? Is this the same as expressing these amounts in base five? Explain.
6. If a light on means 1 and a light off means 0, how would we show any number of ten or fewer digits in base 2 with a panel of ten lights in a row? Draw a picture.
7. Convert to base ten: 75428, 75429, 754211, 211223, 211224, 211225, 211226, 211227, 211228, 5637, 5638, 5639, 56312.
8. Convert the base ten numerals 689, 4955 and 201 to numerals in each of the bases 2, 3, 4, 5, 6, 7, 8 and 9.
9. Convert 512 into yd, ft and in. Convert 39 ft into yd, ft and in. Convert 47 tsp into gal, qt, cup, Tbsp and tsp. Convert 79 Tbsp into gal, qt, cup, Tbsp and tsp.
10. Do addition, subtraction, multiplication and division of two and three digit numbers in base 10, 2, 3, 5 and 6. See Drill 1 handout.
11. Cut out an equilateral triangle and label the vertices 1,2 and 3, in the clockwise order. Write these in heavy enough ink that it can be seen from both sides (or write them on both sides, same number on each vertex). Let R denote clockwise rotation about a fixed point O on your desk top. Let M denote reflection through the line L through O making a 30 degree angle with the horizontal, sloping upward from left to right. Put the triangle on your desk top with its center at the point O and with vertex 1 at the top. Apply R and then M (written MR) to the triangle. What is the configuration of the triangle? Apply RM (that is, apply M and then R) to the the original triangle. What is the configuration of the triangle? Does MR = RM?
12. Explain what needs to be done and then do it:
a). I have $8 and $12, how much do I have?
b). I have $7 and need $20. How much am I short?
c). I have $9, you have $15, how much more do you have than I?
d). Five of us each has $9, how much do we have?
e). I have $43, and each of my 5 children takes $8, how much do I have left?
f). If oranges are 2 for $1, how many can I buy with $4.
g). If oranges are 2 for $1, how much do 11 cost?
h). If I plant 6 rows of trees with 7 trees per row, how many trees will I plant?
i). If I plant 6 rows of trees with 1 tree in the first row, 2 trees in the second row, 3 trees in the third row and so on to 6 trees in the sixth row, how many trees will I plant? What is the answer if I went all the way up to 95 rows, with 95 trees in the 95th row?
13. The temperature rises by 8 degrees Monday, drops by 5 degrees Tuesday, drops 3 degrees Wednesday, rises 2 degrees Thursday and drops 7 degrees Friday. What is the average temperature change for the week? What is the total change. Do the problem using integers, positive for rises and negative for drops. Explain how you do it.
14. Use integers and a number line with East the positive direction.
a). I walk 2 miles east while you walk 3 miles east. How far apart are we at the end of the hour?
b). In the second hour I walk 2 miles east while you walk 3 miles west. Now how far apart are we?
c). In the third hour I walk -2 miles and you walk twice as far in the same direction. Now where are we?
d). In the fourth hour I walk -2 miles and you walk twice as far in the opposite direction. Where are we?
15. Find the prime factorizations of 1 through 100 as well as of 49500, 12500, 38712 and 10083. How many factors does each number have?
16. Observe that 2 + 1 = 3 is prime, 2 x 3 + 1 = 7 is prime and 2 x 3 x 5 + 1 = 31 is prime. Is this a way of producing an unlimited number of primes? Are 2 x 3 x 5 x 7 + 1 = 211 and 2 x 3 x 5 x 7 x 11 + 1 = 2311 and 2 x 3 x 5 x 7 x 11 x 13 + 1 = 30031 prime? Check them to see.
17. Find gcd(28,82), gcd(12,50,60), gcd(99,187,341) and gcd(5,8). Find lcm(28,82), lcm(12,50,60), lcm(99,187,341) and lcm(5,8).
18. Make the addition and multiplication tables for mod 7 and mod 12 arithmetic. Find the multiplicative inverse of every nonzero element in mod 7. Is it possible to do this in mod 12? Which nonzero numbers don't have a multiplicative inverse mod 12? What's the rule. Which nonzero numbers don't have a multiplicative inverse mod 100? What numbers have multiplicative inverses mod 47?
19. Solve the equation 2x + 1 = 4 in mod 7 arithmetic. Can this equation be solved in mod 12 arithmetic? Explain.