Decimal

Also called: base 10, base-10, denary, decimal number system

The everyday base-10 number system, with digits 0 to 9 and place weights 1, 10, 100, 1000 and so on.

Decimal is the number system people use every day. It has ten digits, 0 to 9, and each place is worth ten times the place to its right: ones, tens, hundreds, thousands. As powers, those weights are 10⁰, 10¹, 10², 10³.

In digital logic, decimal is the language you check answers in. Hardware works in binary, but people reason in base 10, so many exercises end with "what is that in decimal?". The method is always the same: multiply each digit by its weight and add.

Everything you already know about decimal carries over to other bases once you swap 10 for the new radix:

a column is full when it reaches the base, and it carries 1 to the left
adding a 0 on the right multiplies by the base
adding a 0 on the left changes nothing

When a number might be read in another base, write the subscript: 10₁₀ is ten, while 10₂ is two.

Worked examples

Example

472, digit by digit

The value of a decimal number is the sum of digit × weight. Every conversion to decimal in this course uses the same sum.

1.
2 is in position 0: 2 × 1 = 2.
2.
7 is in position 1: 7 × 10 = 70.
3.
4 is in position 2: 4 × 100 = 400.
4.
Total: 400 + 70 + 2 = 472.

Example

Decimal and binary side by side

13₁₀ = 1 × 10 + 3. The same amount in binary is 1101₂ = 8 + 4 + 1. Different weights, different digits, same thirteen.

Common mistakes

Assuming a number without a subscript is always decimal. In a hex question, 36 means 36₁₆ = 54.
Mixing up digits and bits: 100 has 3 decimal digits but needs 7 bits.

Practice Decimal

Interactive questions with instant feedback and a worked solution for every wrong answer.

Number Systems lesson

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Decimal is taught in Number Systems.

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