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An online base calculator with steps lets you convert numbers from one base to another base. Not only this, but this free base number calculator will let you apply various arithmetic operations on a couple of numbers in a certain number base system.

So it's time to move ahead and discuss how you could carry out the calculations and conversions among various number systems.

Stay Focused!

In the light of mathematical analysis:

**“A special system that allows numbers to be represented in various notations is known as the number system”**

Following are the types of the number system among which conversions can be carried out by using our best base calculator with steps. Let’s discuss these one by one!

**Binary System:**

- This system has a base value of 2
- It uses only a couple of numbers 0 and 1 to represent the numbers
- The computer system understand only binary numbers as inputs in order to operate properly
- Moreover, if you are interested in knowing more about binary notations, you can use our Binary calculator absolutely for free

**Decimal System:**

- The particular system bears a base number of 10
- It uses ten numerals to display numbers that lie within the range of 0 to 9
- Usually, the numbers to the right of the decimal point will be considered as unit, tenth, hundredth, and so on. You can also verify it by using our online find the adding bases calculator in moments

**Octal System:**

- The base number used to display numbers in this system is 8 since it uses the numbers 0 to 7
- The interesting fact about this system is that that its conversion to decimal system is the same as conversion from decimal to the octal system
- Moreover, if you want to make only conversion to and from this base system, you can use our following converters absolutely for free:

Binary to octal converter Decimal to octal converter Hexadecimal to octal converter

**Hexadecimal System:**

- The base number for the system is 16 as it uses sixteen numbers (0-9 decimal) and upto 15.

The following table can be seen to clarify your mind concept regarding the discussion being carried out.

Decimal base 10 |
Hex base 16 |

0 |
0 |

1 |
1 |

2 |
2 |

3 |
3 |

4 |
4 |

5 |
5 |

6 |
6 |

7 |
7 |

8 |
8 |

9 |
9 |

10 |
A |

11 |
B |

12 |
C |

13 |
D |

14 |
E |

15 |
F |

You can also perform operations on hexadecimal numerals and get detailed results by using our number base calculator. Moreover, if you want to get a detailed map view of the hexadecimal system, you may also use our free hexadecimal calculator to do so.

You may get a thorough understanding of various system values of a number by having a look at the following table:

base 2 |
base 3 |
base 4 |
base 5 |
base 6 |
base 7 |
base 8 |
base 9 |
base 10 |
base 11 |
base 12 |
base 13 |
base 14 |
base 15 |
base 16 |
base 17 |
base 18 |
base 19 |
base 20 |
base 21 |
base 22 |
base 23 |
base 24 |
base 25 |
base 26 |
base 27 |
base 28 |
base 29 |
base 30 |
base 31 |
base 32 |
base 33 |
base 34 |
base 35 |
base 36 |

0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |

1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |

10 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |

11 |
10 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |

100 |
11 |
10 |
4 |
4 |
4 |
4 |
4 |
4 |
4 |
4 |
4 |
4 |
4 |
4 |
4 |
4 |
4 |
4 |
4 |
4 |
4 |
4 |
4 |
4 |
4 |
4 |
4 |
4 |
4 |
4 |
4 |
4 |
4 |
4 |

101 |
12 |
11 |
10 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |

110 |
20 |
12 |
11 |
10 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |

111 |
21 |
13 |
12 |
11 |
10 |
7 |
7 |
7 |
7 |
7 |
7 |
7 |
7 |
7 |
7 |
7 |
7 |
7 |
7 |
7 |
7 |
7 |
7 |
7 |
7 |
7 |
7 |
7 |
7 |
7 |
7 |
7 |
7 |
7 |

1000 |
22 |
20 |
13 |
12 |
11 |
10 |
8 |
8 |
8 |
8 |
8 |
8 |
8 |
8 |
8 |
8 |
8 |
8 |
8 |
8 |
8 |
8 |
8 |
8 |
8 |
8 |
8 |
8 |
8 |
8 |
8 |
8 |
8 |
8 |

1001 |
100 |
21 |
14 |
13 |
12 |
11 |
10 |
9 |
9 |
9 |
9 |
9 |
9 |
9 |
9 |
9 |
9 |
9 |
9 |
9 |
9 |
9 |
9 |
9 |
9 |
9 |
9 |
9 |
9 |
9 |
9 |
9 |
9 |
9 |

1010 |
101 |
22 |
20 |
14 |
13 |
12 |
11 |
10 |
A |
A |
A |
A |
A |
A |
A |
A |
A |
A |
A |
A |
A |
A |
A |
A |
A |
A |
A |
A |
A |
A |
A |
A |
A |
A |

1011 |
102 |
23 |
21 |
15 |
14 |
13 |
12 |
11 |
10 |
B |
B |
B |
B |
B |
B |
B |
B |
B |
B |
B |
B |
B |
B |
B |
B |
B |
B |
B |
B |
B |
B |
B |
B |
B |

1100 |
110 |
30 |
22 |
20 |
15 |
14 |
13 |
12 |
11 |
10 |
C |
C |
C |
C |
C |
C |
C |
C |
C |
C |
C |
C |
C |
C |
C |
C |
C |
C |
C |
C |
C |
C |
C |
C |

1101 |
111 |
31 |
23 |
21 |
16 |
15 |
14 |
13 |
12 |
11 |
10 |
D |
D |
D |
D |
D |
D |
D |
D |
D |
D |
D |
D |
D |
D |
D |
D |
D |
D |
D |
D |
D |
D |
D |

1110 |
112 |
32 |
24 |
22 |
20 |
16 |
15 |
14 |
13 |
12 |
11 |
10 |
E |
E |
E |
E |
E |
E |
E |
E |
E |
E |
E |
E |
E |
E |
E |
E |
E |
E |
E |
E |
E |
E |

1111 |
120 |
33 |
30 |
23 |
21 |
17 |
16 |
15 |
14 |
13 |
12 |
11 |
10 |
F |
F |
F |
F |
F |
F |
F |
F |
F |
F |
F |
F |
F |
F |
F |
F |
F |
F |
F |
F |
F |

Moreover, adding bases as mentioned in the above table can be carried out by using this free of cost base addition calculator.

Let’s move on resolving a few example problems that will let you understand the concept map in depth properly!

**Example:**

Convert the octal number\(\left(123\right)\) to the equivalent decimal number.

**Solution:**

Here we have:

$$ \left(123\right)_8= \left(1x8^{2}\right)+\left(2x8^{1}\right)+\left(3x8^{0}\right) $$

$$ \left(123\right)_8= 16 + 16 + 3 $$

$$ \left(123\right)_8= \left(35\right) $$

Which is our required answer.

**Example:**

Add the following binary numbers:

$$ \left(101\right)_{2} \hspace{0.25in} \left(101\right)_{2} $$

**Solution:**

Here we are having:

(If the value become equal to or greater than 6, then we need to divide to get the remainder to down and take the carry to the next column for addition)

(1 0 1)_{2} + (341)_{2} (442)_{2}

Which is required output against the input provided. For keeping you away from the complexity involved, keep using this find the base calculator absolutely for free.

Now you could instantly apply various operations on numbers bearing different bases and make conversions among base systems with this convert base calculator. Let's find out how it actually works?

**Input:**

- From the top drop-down list, select whether you want to use calculator or converter mode

**If you select calculator mode:**

- Select the number base from the second list
- Now put the numbers in their designated fields
- Select the operation as well
- Hit the calculate button

**If you select the converter mode:**

- Write down the number in the asked field
- Now select the base from which you want to convert the number
- Repeat the same possess for the base in which you wish to convert
- At last, hit the calculate button

**Output:**

The best solve for base calculator does the following calculations for you in a blink of eye:

- Apply different arithmetic operations on the numbers in different base systems
- Convert the numbers from one base system to the other

The corresponding number in the binary system against octal 12 is 1 that you can also cross check by this base calculator.

Different types of numbers are enlisted as follows:

- Prime numbers
- Composite numbers
- Whole numbers
- Integers
- Natural numbers
- Complex numbers

Ordinal numbers are those that are used to define the position of something.

**Example:**

If there are 2 floors of a house, then the first floor will be represented by the 1st floor and the second one will be represented by the 2nd floor.

Number systems have great importance as they allow computer systems and such architectures to be encrypted from hacking issues. And this is why we have introduced this free different base calculator so that students may find it easy to perform various calculations and tackle the complications involved among number system conversions.

From the source of Wikipedia: Numeral system, Main numeral systems, Positional systems, Generalized variable-length integers

From the source of Khan Academy: Binary numbers, Binary numbers, Converting decimal to binary, Patterns

From the source of Lumen Learning: Binary, Octal, and Hexadecimal, Number Systems

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