Number base definition conversion methods and examples
A number base (also known as base) for short is a numeral system that tells us about the unique or different symbols and notations that can be used to represent a value.
For example, the base 2 number system tells that there are only 2 unique notations 0 and 1 to represent the value.
The most commonly used number base is base 10, also known as the decimal number system. The decimal number system uses ten different notations which are the digit 0-9 to represent a value Bases can be either positive, negative, 0, complex, or non-integer. The most frequently used bases are base 2 and base 16. They are also used for calculating and are known as binary, and hexadecimal respectively.
What is a Base Number?
A base number is a number raised to the power that represents the number of units of a number system. For example, the base number of the binary number system is 2.
For Example,
yx
Here, y is a base number.
Base 2 Number System
In Mathematics, the base 2 number system, also known as the binary number system uses 2 as the base and therefore requires only two digits i.e. 0 and 1 to represent any value, rather than 10 different symbols required in the decimal number system. The numbers from 0 to 10 in the binary number system are represented as “.” .The base 2 number system is widely used in Mathematics and Computer Science as bits are easy to create using physically logic gates (the logic gates are either open or closed meaning 0 or 1).
Counting in Different Bases
Counting in different bases substitutes the base 10 with a different bases. We often use Base 10. It is our decimal number system. It has 10 digits.
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
We count numbers with base 10 as shown below:
Let us Understand How to do Counting in Different Base
(Base 2) Binary Number System Has Only 2 Digits: 0 and 1
We count the base 2 like shown below:
(Base 3) Ternary Number System Has 3 Digits: 0,1, and 2
We count numbers with base 3 as shown below:
(Base 4) Quaternary Number System Has 4 Digits: 0, 1, 2, and 3
We count numbers with base 4 as shown below:
(Base 5) Quinary Number System Has 5 Digits: 0, 1, 2, 3, and 4
We count numbers with base 5 as shown below:
(Base 6) Senary Number System Has 6 Digits: 0, 1, 2, 3, 4, and 5
We count numbers with base 6 as shown below:
(Base 7) Septenary Number System Has 7 Digits: 0, 1, 2, 3, 4, 5, and 6
We count numbers with base 7 as shown below:
(Base 8) Octal Number System Has 8 Digits: 0, 1, 2, 3, 4, 5, 6, and 7
We count numbers with base 8 as shown below:
Nonary (Base 9) Number System Has 9 Digits: 0, 1, 2, 3, 4, 5, 6, 7, and 8
We count numbers with base 9 as shown below:
(Base 10) Decimal Number System Has 10 Digits: 0, 1, 2, 3, 4, 5, 6, 7, 9, and 10
We count numbers with base 10 as shown below:
Facts to Remember
In the number system, base, also known as radix, is the number of different digits or combinations of digits and letters that the number system uses to represent numbers.
FAQs on Understanding Bases in Mathematics
1. What are bases in mathematics?
A base in mathematics is the number of unique digits, including zero, used to represent numbers in a positional number system. For example:
- Base 10 (decimal) uses digits 0–9.
- Base 2 (binary) uses digits 0 and 1.
- Base 8 (octal) uses digits 0–7.
- Base 16 (hexadecimal) uses digits 0–9 and A–F.
2. How do you convert a number from base 10 to another base?
To convert a decimal (base 10) number to another base, repeatedly divide by the new base and record the remainders. Steps:
- Divide the number by the new base.
- Write down the remainder.
- Divide the quotient again by the base.
- Repeat until the quotient is 0.
- Read the remainders from bottom to top.
3. How do you convert a number from another base to base 10?
To convert a number from any base to base 10, expand it using powers of the base and add the results. Example: Convert 1011₂ to decimal:
- 1 × 2³ = 8
- 0 × 2² = 0
- 1 × 2¹ = 2
- 1 × 2⁰ = 1
4. What is the difference between binary, octal, decimal, and hexadecimal?
The difference between these number systems is their base value and the digits they use.
- Binary (base 2): digits 0–1
- Octal (base 8): digits 0–7
- Decimal (base 10): digits 0–9
- Hexadecimal (base 16): digits 0–9 and A–F
5. How do you add numbers in different bases?
To add numbers in a given base, follow normal addition rules but carry over when the sum equals or exceeds the base. Example in base 2:
- 1 + 1 = 10₂ (write 0, carry 1)
- 101
- + 011
- = 1000₂
6. What is place value in a number system?
Place value in a number system is determined by powers of the base. In base b, each digit represents a multiple of b raised to a power. For example, in 345₁₀:
- 3 × 10²
- 4 × 10¹
- 5 × 10⁰
7. How do you write numbers in base notation?
Numbers in different bases are written with the base as a subscript. For example:
- 101₂ means base 2
- 17₈ means base 8
- 25₁₀ means base 10
- 1A₁₆ means base 16
8. Can you give an example of converting binary to hexadecimal?
To convert binary to hexadecimal, group binary digits in sets of four and convert each group. Example: 10101100₂:
- Group: 1010 1100
- 1010 = A (10 in decimal)
- 1100 = C (12 in decimal)
9. Why are different number bases used in computing?
Different number bases are used in computing because digital systems operate using binary (base 2). Computers use binary since electronic circuits have two states (on/off).
- Binary is used internally.
- Hexadecimal simplifies reading long binary strings.
- Octal is sometimes used for compact representation.
10. What are common mistakes when working with bases?
Common mistakes when working with number bases include using invalid digits and forgetting place values. Key errors:
- Using digits not allowed in the base (e.g., 2 in binary).
- Forgetting to carry correctly during addition.
- Not using powers of the base when converting to decimal.
- Mixing up base notation.
