The binary, or Base 2, number system uses two digits to express all numerical quantities. The only digits used in the binary number system are 0 and 1. An example of a binary number is 1001110101000110100101.

One important thing to keep in mind is the role of the digit 0. Every number system uses the digit 0. However, note that whenever the digit 0 appears on the left side of a string of digits, it can be removed without changing the string value. For example, in Base 10, 02947 equals 2947. In Base 2, 0001001101 equals 1001101. Sometimes people include 0s on the left side of a number to emphasize "places" that would otherwise not be represented.

Another important concept when working with binary numbers is the powers of numbers. 2⁰ and 2³ are examples of numbers represented by powers. To describe these examples, say "two to the zero" and "two to the three". For example, 2⁰ = 1, 2¹ = 2, 2² = 2 x 2 = 4, 2³ = 2 x 2 x 2 = 8, and so on. Obviously, there is a pattern. The power is the number of 2s that need to be multiplied together. A common mistake is to confuse taking powers with simple multiplication. For example, 2⁴ is not equal to 2 x 4 = 8, instead it is equal to 2 x 2 x 2 x 2 = 16.

In base 10, powers of ten are used. For example, 23605 in base 10 means 2 x 10000 + 3 x 1000 + 6 x 100 + 0 x 10 + 5 x 1.

Note that 10⁰ = 1, 10¹ = 10, 10² = 100, 10³ = 1000, and 10⁴ = 10000.

Caution: Although 0 x 10 = 0, do not leave it out of the above equation. If it is left out, the base 10 places all shift to the right giving the number 2365 = 2 x 1000 + 3 x 100 + 6 x 10 + 5 x 1 instead of 23605. Although a 0 within a number should never be ignored, ignoring or adding 0s to the beginning of numbers has no effect on their value. For instance, 23605 can be expressed as 0023605.

It is useful to think in terms of powers of 10 (10⁰, 10¹, 10², and so on) in relation to a decimal number. When focusing on the actual value of a decimal number, use the expanded form of the powers (1, 10, 100, and so on). It helps to keep track by using tables. In Figure , see the Base 10 number 23605 as it relates to the powers of 10.

Binary
The same method is used with binary numbers and powers of 2. Look at the binary number 10010001. The table in Figure can be used to convert the binary number 10010001 into decimal as follows:

10010001 = 1 x 128 + 0 x 64 + 0 x 32 + 1 x 16 + 0 x 8 + 0 x 4 + 0 x 2 + 1 x 1 = 128 + 16 + 1 = 145

This is one of the ways to convert a binary number into a decimal number. As an exercise, draw a similar table to convert the binary number 11111001 to the decimal number 249. Although the table method is an efficient way to convert binary numbers to decimal, there are other faster methods that can be used.

Lab Activity  (PDF, 29 KB)   In this lab, students will be able to identify the places in binary and decimal numbers and know the value of each. Also, students will work with powers of 10 and relate to decimal places as well as work with powers of 2 and relate to binary places.