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333 LearnersLast updated on November 15, 2025
Number Systems
The number system is a tool we use to represent numbers. Based on the set of digits we have different number systems. Number systems are used to represent quantities, measurements, perform calculations, and so on.
What is a Number System in Math?
The number system is a standardized method used to express numbers using symbols or digits. We categorize numbers as whole numbers, natural numbers, rational numbers, irrational numbers, etc. We use number systems to perform basic arithmetic operations like addition, subtraction, multiplication and division.
History of Number Systems
Since ancient times, numbers have been an essential part of our everyday life. Early people used stones, bones, and tally marks for counting. As life grew more complex, different number systems were developed, such as Egyptian numerals, based on powers of ten; Mesopotamian cuneiform; Babylonian sexagesimal (base 60); and Roman numerals, using letters like I, V, X, L, C, D, and M.
The Arabic Number System, also known as the Decimal System, is the modern system we use today. It is based on the powers of 10 and follows the concept of place value. With the growth of technology, other systems like Binary, Octal, and Hexadecimal systems were also developed.
Interestingly, the idea of number systems also extends beyond mathematics, for example, the Nashville Number System, used by musicians to represent musical chords with numbers instead of letters. It shows how the idea of number systems extends beyond mathematics into fields such as music and the creative arts.
Classification of Number System
The number system is classified into various types depending on their properties. In this section, we will be discussing the four main types of number system.
- Binary Number System
- Decimal Number System
- Octal Number System
- Hexadecimal Number System
Binary Number System
- The base of this number system is 2
- The binary system uses only two digits, 0 and 1. Here, no other numbers are used.
- Here, the digits 0 and 1 are known as binary numbers
- It is used in computer systems and electronic devices.
Decimal Number System
- The base of this number is 10
- Here, the numbers 0 to 9 are used to create numbers
- The place value is calculated from right to left
Octal Number System
- The base of this number system is 8
- To create a number here, 8 digits are used is 0 to 7
- The decimal system can be converted to the octal number system.
Hexadecimal Number System
- The alphabets A to F are used to represent the numbers from 10 to 15
- The base of this number system is 16.
- Here, numbers and alphabets are used from 0 to 9 and from A to F.
Types of Number Systems
Numbers can be classified into positional and non-positional systems, we base it on the representation of the numbers. When we express it, we classify it further into a standard form or expanded form.
Positional Number System:
The positional number system also known as the weighted number system is based on the weight or value of the digits according to the positions of the numbers. We relate the number that is right next to it. Decimal, binary, octal, and hexadecimal number systems are some of the few types of positional number systems.
Example: 14 can be \(1 ×10 + 4 × 1 = 10 + 4 = 14\)
Non-Positional Number Systems
When each number has its own value, we call that a non-positional number system. The value of the number doesn’t change with the place value because numbers have no relation to their place values. One such example of non-positional number systems are Roman numerals.
Standard Form:
Standard form is a way of expressing numbers in the form of numbers. In standard form, 5234 is written as \(5.234 × 10^3\)
Expanded Form:
Here, we express the number using the place value of the number. In expanded form, 5234 is written as \(5000 + 200 + 30 + 4. \)
Properties of Number Systems
The properties of numbers are based on their characteristics, the basic properties are:
- Commutative Property
- Associative Property
- Distributive Property
- Identity Property
- Addition
- Subtraction
- Multiplication
- Division
Commutative Property: When we perform multiplication and addition, the order of the numbers does not affect the result.
Example: In addition: \(5 + 2 = 2 + 5 = 7.\)
In multiplication: \(5 × 2 = 2 × 5 = 10.\)
Associative Property: When we add or multiply three or more numbers, the order of the groups does not affect the result.
Example: In addition: \((2 + 5) + 3 = (3 + 5) + 2 = 10.\)
In multiplication: \((3 × 5) × 2 = (2 × 5) × 3 = 30.\)
Distributive Property: If we add two or more addends, multiplying the sum by a number will give the same result as multiplying the number by each addend and then adding the products
Example: \(3 × (4 + 5) = (3 × 4) + (3 × 5) = 27.\)
Identity Property: When we multiply 1 and add 0 to any number, the product and sum will be the same.
Example: In addition: \(4 + 0 = 4\)
In multiplication: \(5 × 1 = 5\)
Importance of Number Systems for Students
The number system is used in our daily life. In this section, let's learn more about the number system and how it is helpful for students.
- Do basic calculations using mathematical operations
- Measure the distance or qualities of the object
- Solve the arithmetic progressions and equations
- Help to understand the number value
- Understanding the currency exchange rate
Tips and Tricks to Master Number Systems
Learning a number system can help students to understand the basic characteristics of numbers and number systems. Now let’s discuss a few tips and tricks to master the number system.
Understanding the concept of base: Each number system will have a different base, this makes the numbers of all the number systems unique. In the binary system, the base is 2, in the decimal system the base is 10, in the octal system the base is 8, and in hexadecimal the base is 16.
Understand the pattern: The number system follows different patterns, and it makes it unique. So, understanding the pattern will help kids to master the number system.
Using online tools and calculators: To make conversion of numbers system easier, students can use online tools and calculators.
Use Daily Life Examples: Show the numbers using things around, like counting pencils, fruits, or steps.
Draw and Explain: Use number lines and charts to show how numbers increase, decrease, or change from positive to negative.
Teach Step by Step: When teaching the real numbers, explain how they include Rational Numbers, fractions, decimals, and Irrational Numbers. Show step-by-step conversions between Decimal, Binary, Octal, and Hexadecimal systems.
Make Learning Fun: Use games, puzzles, or short quizzes to help students remember easily.
Show Real-Life Use: Explain how computers, calculators, and clocks use different number systems in daily life.
Common Mistakes and How to Avoid Them in Number Systems
Learning a number system can be complicated, as each number system has rules of their own. In this section, let’s discuss some common mistakes and how to avoid them to master the number system.
Mistake 1
Confusing with base
The number system is unique as it has a different base, so students will get confused easily. To avoid this error, students should learn and understand the base of each number system. For Binary the base is 2, octal the base is 8, decimal the base is 10, and hexadecimal is 16 .
Mistake 2
Errors in conversion
When converting from one form to another, students should understand the base of each number system before conversion.
Mistake 3
Confusing with hexadecimal digits
The hexadecimal number system is the only number system with numbers and alphabets, so students tend to confuse it. The values of letters A to F can be miscalculated, so students memorize the values. That is A = 10, B = 11, C = 12, D = 13, E = 14, F = 15.
Mistake 4
Misunderstanding the place value in positional system
Each digit’s position in numbers has significance. So students should expand the numbers to understand the place value.
Mistake 5
Using invalid digits
The number system uses a set of numbers. We use 0 and 1 for binary, in octal 0 - 7, and in decimal we use 0 - 9. Students sometimes confuse the digits, to avoid it students should learn the digits for each system.
Real-World Applications of Number Systems
To represent the qualities or values, we use numbers. For counting, measuring, coding, and so on we use number systems.
- In computers: To process data we use the binary number system, in some cases hexadecimal and octal number systems are used as well.
- We use hexadecimal and octal number systems to make codes easier to read and debug.
- In cryptography, binary and hexadecimal are widely used in encryption algorithms for data security.
- In financial transactions, we use binary and decimal numbers systems in banking and stock market algorithms to process transactions.
Solved Examples on Number Systems
Problem 1
Convert the decimal number 25 to binary.
25 in binary is, 11001.
Explanation
To convert decimal to binary:
Step 1: Divide 25 by 2 where the quotient is 12 and the remainder is 1.
Step 2: Divide 12 by 2 where the quotient is 6 and the remainder is 0.
Step 3: Divide 6 by 2 where the quotient is 3 and the remainder is 0.
Step 4: Divide 3 by 2 where the quotient is 1 and the remainder is 1.
Step 5: Finally divide 1 by 2 where the quotient is 0 and the remainder is 1.
Now read the remainders from bottom to top.
So 25 in binary form = 11001.
Problem 2
Convert the binary number 1011 into decimal.
1011 in decimal is 11.
Explanation
The rightmost digit of the binary number is multiplied with the corresponding power of 2.
The process is repeated with all the digits, and the product in all the steps is added to get the decimal number.
Problem 3
Subtract the hexadecimal number 2A from 3F.
The difference between 3F and 2A is F.
Explanation
Here, F is 15 and A is 10. We subtract the values separately, that is, we first subtract the numbers, 3 - 2 = 1.
Then we subtract F and A, F - A = 15 - 10 = 5.
Now combine 1 and 5. So the result we get is 15.
When we convert 15 into hexadecimal, the number is F.
Problem 4
Multiply the binary 11 and 101.
The product is, 1111.
Explanation
The numbers are arranged in order, just like decimal multiplication.
That is, \(101 × 11 = 1111. \)
Problem 5
Find the sum of 37 and 42 in octal numbers.
The sum of 37 and 42 is 101.
Explanation
Here the number is added based on the place value that is 7 + 2 and 3 + 4.
\(7 + 2 = 9\) (units).
\(3 + 4 = 7\) (eights).
Combining the result that is 79. As there is no 9 in octal numbers, \(9 = 1 × 8 + 1\). So the final result is 101.
FAQs on Number Systems
1.What are the types of number systems?
2.What is z in math?
3.What is a number system?
4.Which number system is based on numbers 0 and 1?
5.What is the difference between decimal and hexadecimal systems?
6.What should parents know about the Number System and how can they explain it to their child?
7.How can parents teach their child about the Real Number System?
Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
