Solutions for Session 3, Part C

See solutions for Problems: C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8

 Problem C1 123four = (1 • 42) + (2 • 41) + (3 • 40) = 16 + 8 + 3 = 27. The illustration demonstrates the number 27 divided into groups of 42, 41, and 40. 1.23four = (1 • 40) + (2 • 4-1) + (3 • 4-2) = 1 + 2/4 + 3/16 = 1 11/16.

Problem C2

 a. 0.1four is equal to 1/4 in base ten. 0.2four is twice 0.1four, so it is 2/4 = 1/2. 0.3fouris three times 0.1four, so it is 3/4 in base ten. b. 0.01four is equal to 1/16 in base ten. 0.02four is twice 0.01four, so it is 2/16 = 1/8. 0.03four is three times 0.01four, so it is 3/16 in base ten. c. 0.11four = 0.1four + 0.01four. This sum is 1/4 + 1/16 = 5/16. Similarly, 0.12four = 1/4 + 2/16 = 6/16 = 3/8, and 0.13four = 1/4 + 3/16 = 7/16.

Problem C3

 a. 1/2 is equivalent to 2/4, so as a base four decimal, it would be written as 0.2four. b. 5/8 = 4/8 + 1/8 = 2/4 + 2/16 = 0.22 in base four. c. 7/8 = 6/8 + 1/8 = 3/4 + 2/16 = 0.32 in base four. d. 1/64 = 0.001 in base four.

Problem C4

 a. Since the digits in base four are 0, 1, 2, and 3, the last digit before "rolling over" to the next place value is 3. The number just before 100 is 33 (. . ., 30, 31, 32, 33, 100). b. One more than 133 is 200. Adding 1 to 133 gives us one 16, three 4s, and four 1s. Four 1s equals one 4, so this gives us one 16, four 4s, and zero 1s. Four 4s equals one 16, so we now have two 16s, zero 4s, and zero 1s -- 200. c. The greatest three-digit number is 333. Just before 333 is 332, and just after 333 is 1000.

Problem C5

 a. The count goes 2, 10, 12, 20, 22, 30. b. Even numbers in base four end in 0 or 2. This can be seen by the previous question; any number in the list formed from counting by twos is even. c. The count goes 3, 12, 21, 30.

Problem C6

 a. 33four + 11four =110four b. 123four + 22four = 211four c. 223four - 131four = 32four d. 112four - 33four = 13four