Session 3, Part C:
Place-Value Representation in Base Ten
and Base Four
(40 minutes)

In This Part: Examining Base Four | Operations in Base Four

In Part C, we shift our focus to the base four number system. You will learn how to interpret whole numbers, common fractions, and decimals using this system.

In base ten, 123 means (1 • 100) + (2 • 10) + (3 • 1), and 1.23 means (1 • 1) + (2 • [1/10]) + (3 • [1/100]). Or, to put it another way:

 123ten= (1 • 102) + (2 • 101) + (3 • 100) and 1.23ten= (1 • 100) + (2 • 10-1) + (3 • 10-2)

We can represent each of the place values in the base ten number 123 with pieces of 100 units (102), 10 units (101), and one unit (100). They are called flats, longs, and units respectively.

The base four number system uses these place values:

 44 43 42 41 40 4-1 4-2 256 64 16 4 1 1/4 1/16

 So in base four, 123 means: and 1.23 means: (1 • 16) + (2 • 4) + (3 • 1), (1 • 1) + (2 • [1/4]) + (3 • [1/16]), or or 123four = (1 • 42) + (2 • 41) + (3 • 40), 1.23four = (1 • 40) + (2 • 4-1) + (3 • 4-2).

We can represent each of the place values in the base four number 123 with pieces of 16 units (42), four units (41), and one unit (40). They are called flats, longs, and units respectively.

Problem C1

Write the base four numbers 123four and 1.23four in expanded notation and complete the base ten value of the number.

Problem C2

Find the base ten fractions represented by the following:

 a. 0.1four, 0.2four, and 0.3four b. 0.01four, 0.02four, and 0.03four c. 0.11four, 0.12four,and 0.13four

 What number is represented by "11" in base four? It is not what we would call 11!   Close Tip What number is represented by "11" in base four? It is not what we would call 11!

Problem C3

Find the base four representation for these base ten fractions:

 a. 1/2 b. 5/8 c. 7/8 d. 1/64

Note 2

 Play with these fractions to get them into the desired form x/4 + y/16 + z/64.... Remember that in base four, the face values of x, y, and z can only be digits 0 though 3.   Close Tip Play with these fractions to get them into the desired form x/4 + y/16 + z/64.... Remember that in base four, the face values of x, y, and z can only be digits 0 though 3.

Problem C4

 a. If you were counting in base four, what number would you say just before you said 100? (Read this number as "one-zero-zero," not "one hundred.") b. What number is one more than 133? (Read this number as "one-three-three.")

 Use the base four blocks diagram above.   Close Tip Use the base four blocks diagram above.

 c. What is the greatest three-digit number that can be written in base four? What numbers come just before and just after this number?

 Video Segment In this video segment, Ben and Liz work with manipulatives to represent numbers in base four and solve some arithmetic problems. They realize that they need to move to the next place value in base four. Watch this segment after you've completed Problems C1-C4. Did you find it necessary to think about what each place value means in order to solve these problems in base four? If you are using a VCR, you can find this segment on the session video approximately 16 minutes and 15 seconds after the Annenberg Media logo.

Problem C5

 a. Count by twos to 30four. b. In base four, how can you tell if a number is even?

 Look for a pattern in the results of the first question to help you answer the second.   Close Tip Look for a pattern in the results of the first question to help you answer the second.

 c. Count by threes to 30four.

 Session 3: Index | Notes | Solutions | Video