Introduction to Number Systems and the Decimal System

Get introduced to number systems, and understand their basics using everyday numbers as an example.

We'll cover the following

What is a number system?

Evidence from early civilizations shows that humans have been counting for a long time. Indeed, counting is an essential tool for daily life. So, naturally, humans have developed systems that relate unique symbols to specific values. These are what we call number systems.

What traits should a number system have?

Let’s take an example from early civilization, where an employee and an employer are selling fruits in the market. The employer hands over a certain number of fruits to sell in the market to the employee. The employer has a pile of peach marbles with him, which is exactly equal to the number of fruits he has handed over to his employee.

The employee sells all the fruits and returns to their employer. The employee is expected to bring along another pile of marbles (shown in green color), equivalent to the number of fruits they sold. The employer compares the two piles of marbles by doing one-to-one matching, observing whether they are smaller/larger or if both piles are equal. This was their simplest way of counting in the absence of a number system.

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Now suppose that we want to do this without having both piles in front of us. We want to count one pile, quantify it to a value, and then have someone else tell us the value of the other pile. Then, like before, we want to compare the two values, to determine which is greater, or if they’re equal. Perhaps, in our number system, we also want to learn exactly how much greater one value would be in comparison to the other.

Essentially, we need our system to:

  • Have unique symbols for every value
  • Be consistent and have comparable values
  • Be easily scalable

We will discuss each of these traits in turn. Let’s start with the third.

Scalability

If we want widespread use of our system, we can’t expect everyone to memorize symbols upon symbols. The expectation is that some basic knowledge would allow perfect use of the system. For example, in our everyday number system, all numbers beyond 9 are generated by a combination of the existing numbers.

Consistency and comparability

Scalability is easily possible if our system is consistent. That means that the rules that apply to the first few numbers in our system must also apply to the ones after that and so on.

As we saw, comparability is an essential factor. Two numbers (however large) should be comparable just by looking at their structure and knowing the derivation rules and basics of the system.

Uniqueness

Finally, we have the most important property: unique symbols for every value. Without unique symbols, we can’t have a usable system.

To achieve this, we have a set of n{n} base symbols representing the first n{n} values in our number system. The subsequent values would be combinations of these n{n} symbols, derived by following specific rules. This way, we ensure uniqueness, but with an inherent scalability and consistency. The simplest example is our everyday number system: no two values are represented by the same symbol combination.

The decimal system

We will discuss the idea of using n{n} base symbols to build a number system using the example of our everyday number system: the decimal system.

Base symbols

The decimal system has 10 base symbols, or digits. Thus, it is also called the base-10 system. We learn these symbols, the numbers 0–9, as soon as we begin our schooling. We also learn the rules to construct further numbers: 10 and onwards.

Place value

The rules on how to construct a number in the decimal system (and in all n{n}-base number systems) depend on the number’s place value. This is the value a digit will have according to its position in the sequence of the number.

In the decimal number system, the place value of each digit increases in powers of 10 from right to left. The rightmost digit represents 10010^0 or 11, the next digit to the left represents 10110^1 or 1010, the next one represents 10210^2 or 100100, and so on. This is because the decimal system uses a base of 1010, meaning we have 1010 base symbols (0—9)(0—9) to represent numbers.

Test yourself

Q

(Fill in the blank.) The expansion, in terms of powers of 10 of the number 795 is ______.

A)

77 x 10210^{2} ++ 99 x 10110^{1} ++ 55 x 10010^{0}

B)

55 x 10210^{2} ++ 99 x 10110^{1} ++ 77 x 10010^{0}

C)

55 x 10310^{3} ++ 99 x 10210^{2} ++ 77 x 10110^{1}