Learning Math: Number and Operations
Place Value Part C: PlaceValue Representation in Base Ten and Base Four (40 minutes)
In This Part: Examining 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:
123_{ten}= (1 • 10^{2}) + (2 • 10^{1}) + (3 • 10^{0}) 
and 
1.23_{ten}= (1 • 10^{0}) + (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 (10^{2}), 10 units (10^{1}), and one unit (10^{0}). They are called flats, longs, and units respectively.
The base four number system uses these place values:


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 

123_{four} = (1 • 4^{2}) + (2 • 4^{1}) + (3 • 4^{0}), 
1.23_{four} = (1 • 4^{0}) + (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 (4^{2}), four units (4^{1}), and one unit (4^{0}). They are called flats, longs, and units respectively.
Problem C1
Write the base four numbers 123_{four} and 1.23_{four} in expanded notation and complete the base ten value of the number.
Problem C2
Find the base ten fractions represented by the following:
a. 0.1_{four}, 0.2_{four}, and 0.3_{four
}b. 0.01_{four}, 0.02_{four}, and 0.03_{four
}c. 0.11_{four}, 0.12_{four},and 0.13_{four}
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
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 “onezerozero,” not “one hundred.”)
b. What number is one more than 133? (Read this number as “onethreethree.”)
Use the base four blocks diagram above.
c. What is the greatest threedigit 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 C1C4.
Did you find it necessary to think about what each place value means in order to solve these problems in base four?
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 30_{four}.
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.
c. Count by threes to 30_{four}.
In This Part: Operations in Base Four
The following Interactive Activity provides you with some base four blocks you can use to visualize place value in base four operations. The pieces are called flats, longs, and units, and they represent 4^{2}, 4^{1}, and 4^{0}, respectively. Use the blocks to solve Problem C6.
Remember that there are four digits in base four (0 through 3). For addition problems, every time you have four blocks of the same type (a given place value), you need to trade them (or regroup them) for one block of the nextlarger place value on the left. So, for example, four blocks representing 4^{0} (four units) would be traded for one block representing 4^{1} (one long). Use the regroup button in the activity to do this.
Similarly, for subtraction, every time you don’t have enough blocks in a given place value to do the subtraction, you need to take one larger block from the place value on the left and trade it for four blocks of the nextsmaller place value on the right. So, for example, one block representing 4^{1} (one long) can be traded for four blocks representing 4^{0} (four units). Use the regroup button in the activity to do this. See Note 3 below.
For a noninteractive version of this activity, print and cut out several base four blocks PDF to use with Problem C6.
Problem C6
Complete the following calculations of these base four numbers:
a. 33_{four} + 11_{four
}b. 123_{four} + 22_{four
}c. 223_{four} – 131_{four
}d. 112_{four} – 331_{four}
Video Segment
Why is computing in different bases useful? In this segment, Mr. Glasgow and Mr. Marable explain how today’s computer technology relies on base sixteen numbers in order to compress complex computer operations.
Can you make a prediction about what number systems computers will use in the future?
You can find this segment on the session video approximately 22 minutes and 24 seconds after the Annenberg Media logo. The second part begins approximately 25 minutes and 14 seconds after the Annenberg Media logo.
Notes
Note 2
We will learn more about converting fractions to decimals in Session 7.
Note 3
Think of how the base four blocks help you understand operations with base four numbers. Similarly, contemplate how base ten blocks would have helped you when you learned to add, subtract, multiply, and divide in base ten.
Solutions
Problem C1
123_{four} = (1 • 4^{2}) + (2 • 4^{1}) + (3 • 4^{0}) = 16 + 8 + 3 = 27. The illustration demonstrates the number 27 divided into groups of 4^{2}, 4^{1}, and 4^{0}.
1.23_{four} = (1 • 4^{0}) + (2 • 4^{1}) + (3 • 4^{2}) = 1 + 2/4 + 3/16 = 1 11/16.
Problem C2
a. 0.1_{four} is equal to 1/4 in base ten. 0.2_{four} is twice 0.1_{four}, so it is 2/4 = 1/2. 0.3_{four}is three times 0.1_{four}, so it is 3/4 in base ten.
b. 0.01_{four} is equal to 1/16 in base ten. 0.02_{four} is twice 0.01_{four}, so it is 2/16 = 1/8. 0.03_{four} is three times 0.01_{four}, so it is 3/16 in base ten.
c. 0.11_{four} = 0.1_{four} + 0.01_{four}. This sum is 1/4 + 1/16 = 5/16. Similarly, 0.12_{four} = 1/4 + 2/16 = 6/16 = 3/8, and 0.13_{four}= 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 0.2_{four}.
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 threedigit 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. 33_{four} + 11_{four} =110_{four
}b. 123_{four} + 22_{four} = 211_{four
}c. 223_{four} – 131_{four} = 32_{four
}d. 112_{four} – 33_{four} = 13_{four}