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Decimal to Binary: Definition and Examples | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Decimal to BinaryDecimal to Binary: Definition and ExamplesTable of ContentsConvert Decimal To Binary Definition of Decimal to Binary Conversion

The decimal number system, also known as the Hindu-Arabic number system, is a base-101010 number system that uses 101010 digits: 000, 111, 222, 333, 444, 555, 666, 777, 888, and 999. Each digit in this system has a place value that increases by powers of 101010 from right to left (ones, tens, hundreds, etc.). For example, in the number 245245245, the 555 represents 5×15×15×1, the 444 represents 4×104×104×10, and the 222 represents 2×1022×10^22×102.

The binary number system is a base-222 number system that uses only two digits: 000 and 111. Each digit in a binary number is called a bit. In binary, place values increase by powers of 222 from right to left (ones, twos, fours, etc.). Converting from decimal to binary involves dividing the decimal number repeatedly by 222 and noting down the remainders. When the quotient becomes 000, the binary equivalent is found by writing these remainders in reverse order. For decimal fractions, the fractional part is converted by repeatedly multiplying by 222 and noting the integer parts.

Examples of Decimal to Binary Conversion Example 1: Converting a Simple Decimal to Binary Problem:

Convert 141014_{10}1410​ into binary.

Step-by-step solution:

Step 1, Divide 141414 by 222. The result is quotient 777 and remainder 000.

14÷2=714 ÷ 2 = 714÷2=7 with remainder 000

Step 2, Divide 777 by 222. The result is quotient 333 and remainder 111.

7÷2=37 ÷ 2 = 37÷2=3 with remainder 111

Step 3, Divide 333 by 222. The result is quotient 111 and remainder 111.

3÷2=13 ÷ 2 = 13÷2=1 with remainder 111

Step 4, Divide 111 by 222. The result is quotient 000 and remainder 111.

1÷2=01 ÷ 2 = 01÷2=0 with remainder 111

Step 5, Write down all the remainders from bottom to top (in reverse order).

1410=1110214_{10} = 1110_21410​=11102​ Example 2: Converting a Decimal Fraction to Binary Problem:

Find the binary fraction equivalent of 0.125100.125_{10}0.12510​

Step-by-step solution:

Step 1, Multiply 0.1250.1250.125 by 222. The result is 0.2500.2500.250.

0.125×2=0.2500.125 × 2 = 0.2500.125×2=0.250 Write down the integer part: 000

Step 2, Multiply the fractional part 0.2500.2500.250 by 222. The result is 0.5000.5000.500.

0.250×2=0.5000.250 × 2 = 0.5000.250×2=0.500 Write down the integer part: 000

Step 3, Multiply the fractional part 0.5000.5000.500 by 222. The result is 1.0001.0001.000.

0.500×2=1.0000.500 × 2 = 1.0000.500×2=1.000 Write down the integer part: 111

Step 4, Since the fractional part is now 000, we stop. Write down the integer parts in the same order they were obtained.

0.12510=0.00120.125_{10} = 0.001_20.12510​=0.0012​ Example 3: Converting a Mixed Decimal to Binary Problem:

Convert 10.21010.2_{10}10.210​ to base 222 number system.

Step-by-step solution:

Step 1, Split the number into integer part (101010) and fractional part (0.20.20.2).

Step 2, Convert the integer part 101010 to binary:

10÷2=510 ÷ 2 = 510÷2=5 with remainder 000 5÷2=25 ÷ 2 = 25÷2=2 with remainder 111 2÷2=12 ÷ 2 = 12÷2=1 with remainder 000 1÷2=01 ÷ 2 = 01÷2=0 with remainder 111 Reading the remainders from bottom to top: 1010=1010210_{10} = 1010_21010​=10102​

Step 3, Convert the fractional part 0.20.20.2 to binary:

0.2×2=0.40.2 × 2 = 0.40.2×2=0.4 (integer part: 000) 0.4×2=0.80.4 × 2 = 0.80.4×2=0.8 (integer part: 000) 0.8×2=1.60.8 × 2 = 1.60.8×2=1.6 (integer part: 111) 0.6×2=1.20.6 × 2 = 1.20.6×2=1.2 (integer part: 111)

Step 4, Write down the integer parts in the same order: 001100110011...

Note: This will repeat indefinitely since 0.20.20.2 in decimal doesn't have a finite binary representation. We can approximate: 0.210≈0.001120.2_{10} ≈ 0.0011_20.210​≈0.00112​

Step 5, Combine the integer and fractional parts:

10.210=1010.0011210.2_{10} = 1010.0011_210.210​=1010.00112​ Comments(1)SSoftballPlayerOscarNovember 6, 2025I've been struggling to explain decimal to binary to my students. This page's clear def and examples made it so much easier! Thanks!

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