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Although the number system commonly used for counting and measuring is based on the number 10 and is known as the decimal system, there are counting systems based on other numbers. For example, base-2 and base-60 number systems are also used for counting. Base-2, known as the binary number system, is used in electronic computers and other electrical devices. Time on a clock is partially measured in the base-60 system. Each hour is divided into 60 minutes and each minute is divided into 60 seconds. This entry will introduce the base-10 and base-2 number systems.

Base-10 Number System

Because humans have ten fingers, objects are naturally grouped in tens when counting. Counting a dozen apples with one's fingers consists of counting up to ten and then repeating the count, which results in one 10 plus two 1s. So all numbers in this base-10 system are made from just ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

For example, there are 365 days in a year. In the base-10 number system, the value of each of the three digits 3, 6, and 5 depends on their position. Starting from the left, the 3 stands for 3 hundreds; the 6 stands for 6 tens; the 5 stands for 5 ones, or units. So,

365 = (3 × 100) + (6 × 10) + (5 × 1).

Using exponents, 100 = 10², 10 = 10¹, and 1 = 10⁰. So,

365 = (3 × 10²) + (6 × 10¹) + (5 × 10⁰).

In a similar fashion, 2,030 is expressed as (2 × 10³) + (0 × 10²) + (3 × 10¹) + (0 × 10⁰).

Base-2 Number System

Whereas the base-10 number system is naturally suited to humans, base-2 is suited to computers and other devices that run on electricity. The electric current has two states—on and off. A computer is programmed to compute with groups of two using the binary number system.*

*The word "binary" means "comprised of two."

In base-10, a number is expressed in terms of the sum of multiples of 10: 10⁰, 10¹, 10², and so on. But in base-2, a number is expressed in terms of the sum of multiples of 2: 2⁰, 2¹, 2², and so on. This basically means that objects are grouped in twos. The following example shows how to express four base-10 digits in binary form.

2 (one group of two) (1 × 2¹) + (0 × 2⁰) + 10

3 (one group of two plus one) = (1 × 2¹) + (1 × 2⁰) = 11

4 (two groups of two; same as one group of four) = (1 × 2²) + (0 × 2¹) + (0 × 2⁰) = 100

5 (one group of four plus one) = (1 × 2²) + (0 × 2¹) + (1 × 2⁰) = 101

The binary number 1011, for example, is equal to (1 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰), which in base-10 equals 11. So the binary (base-2) number 1011 and the decimal (base-10) number 1,011 represent totally different values.

SEE ALSO COMPUTERS AND THE BINARY SYSTEM; DECIMALS; POWERS AND EXPONENTS; TIME, MEASUREMENT OF.

Rafiq Ladhani

Bibliography

Amdahl, Kenn, and Jim Loats. Algebra Unplugged. Broomfield, CO: Clearwater Publishing Co., 1995.

Miller, Charles D., Vern E. Heeren, and E. John Hornsby, Jr. Mathematical Ideas, 9th ed. Boston: Addison-Wesley, 2001.