 Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum  MENU          Session 3, Part A:
Base Two Numbers (45 minutes)

In This Part: Base Two | Converting Between Bases | Base Two Numbers in Computing

The number systems we've been looking at so far in this course use 10 digits (0 through 9), and the value of each position in a number is some power of 10 (1; 10; 100; 1,000; etc.). We refer to this number system as base ten. But numbers can also be written in other bases. In base two, for example, we have two digits (0 and 1), and the value of each position in a number is some power of 2.

To interpret numbers in base ten, we must look at each digit and determine the value of that digit according to its place in the number. The convention we use is that each place value, moving from right to left, represents an increasing power of 10: In order to understand the value of a number, we need to consider both its face values and its place values. In 2,342, there are two 2s (the face values), but each of these 2s has a different place value (1,000 and 1). Thus, the value of any number is found by multiplying each face value by its place value and then adding the results.

For example, the value of 234,567 in base ten is

(2 • 105) + (3 • 104) + (4 • 103) + (5 • 102) + (6 • 101) + (7 • 100),

or

(2 • 100,000) + (3 • 10,000) + (4 • 1,000) + (5 • 100) + (6 • 10) + (7 • 1).

Similarly, we can consider numbers that are less than 1. The following two numbers may look similar if we look only at their face values. However, they have different place values, which is evident from the following:

0.02 = (0 • 100) + (0 • 10-1) + (2 • 10-2) = 2 • 10-2

0.002 = (0 • 100) + (0 • 10-1) + (0 • 10-2) + (2 • 10-3) = 2 • 10-3

Thus, 0.02 and 0.002 are two distinct numbers in the base ten system.

Interpreting numbers in base two works the same way, except that the place value of each digit is some power of 2 instead of 10. We determine the value of digits in a base two number in a similar way. Remember, though, we can only use two digits in this system as face values, the digits 0 and 1, and the place values are powers of 2.

Here are base two place values, written as base ten numbers:

 ... 25 24 23 22 21 20 2-1 2-2 ... ... 32 16 8 4 2 1 1/2 1/4 ...

Note that both 100 and 20 equal 1. This and other rules about exponents will be explored further in the next part of this session.

We can also look at a base two number and find its value in base ten. For example, the base ten value of the base two number 101110 is:

(1 • 25) + (0 • 24) + (1 • 23) + (1 • 22) + (1 • 21) + (0 • 20),

or

(1 • 32) + (0 • 16) + (1 • 8) + (1 • 4) + (1 • 2) + (0 • 1) = 46.  Problem A1 Try counting in base two. Explain the patterns you see, and compare them to our base ten system.  Look at what happens to particular place values as you count.   Close Tip Look at what happens to particular place values as you count.   Session 3: Index | Notes | Solutions | Video

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