High School Physics

Decimal System with examples

In this post, we will cover Decimal System with examples. The decimal system is composed of 10 numerals or symbols. These 10 symbols are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9; using these symbols as digits of a number, we can express any quantity.

Decimal System or base-10 system with examples

The decimal system, also called the base-10 system because it has 10 digits, has evolved naturally as a result of the fact that people have 10 fingers. In fact, the word digit is derived from the Latin word for “finger.”

The decimal system is a positional-value system in which the value of a digit depends on its position.

For example, consider the decimal number 453. We know that the digit 4 actually represents 4 hundreds, the 5 represents 5 tens, and the 3 represents 3 units. In essence, the 4 carries the most weight of the three digits; it is referred to as the most significant digit (MSD). The 3 carries the least weight and is called the least significant digit (LSD).

Consider another example, 27.35. This number is actually equal to 2 tens plus 7 units plus 3 tenths plus 5 hundredths, or 2 x 10 + 7 x 1 + 3 x 0.1 + 5 x 0.01. The decimal point is used to separate the integer and fractional parts of the number.

More rigorously, the various positions relative to the decimal point carry weights that can be expressed as powers of 10. This is illustrated in Figure 1, where the number 2745.214 is represented. The decimal point separates the positive powers of 10 from the negative powers. The number 2745.214 is thus equal to

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(2 * 10+3) + (7 * 10+2) + (4 * 101) + (5 * 100)+ (2 * 10-1) + (1 * 10-2) + (4 * 10-3)

figure 1: Decimal position values as powers of 10.
figure 1: Decimal position values as powers of 10.

In general, any number is simply the sum of the products of each digit value and its positional value.

Decimal Counting

When counting in the decimal system, we start with 0 in the unit’s position and take each symbol (digit) in progression until we reach 9. Then we add a 1 to the next higher position and start over with 0 in the first position (see figure 2)

This process continues until the count of 99 is reached. Then we add a 1 to the third position and start over with 0s in the first two positions.

The same pattern is followed continuously as high as we wish to count.

figure 2: Decimal Counting
figure 2: Decimal Counting

It is important to note that in decimal counting, the units position (LSD) changes upward with each step in the count, the tens position changes upward every 10 steps in the count, the hundreds position changes upward every 100 steps in the count, and so on.

Another characteristic of the decimal system is that using only two decimal places, we can count through 102 = 100 different numbers (0 to 99).*

With three places we can count through 1000 numbers (0 to 999), and so on. In general, with N places or digits, we can count through 10N different numbers, starting with and including zero. The largest number will always be 10N – 1

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