Converting Number Systems

The default decimal, Base₁₀, system ideally should be annotated 0, 1₁₀, 2₁₀, 3₁₀, 4₁₀, 5₁₀, 6₁₀, 7₁₀, 8₁₀, 9₁₀, but the subscripts are omitted in everyday use.

The Decimal Base₁₀ system columns

Column Name 10Mils Mils 100Ths 10Ths Ths 100s 10s Units

Base₁₀ Column Value 10⁷ 10⁶ 10⁵ 10⁴ 10³ 10² 10¹ 10⁰

Decimal Column Value 10Mil₁₀ 1Mil.₁₀ 100Th.₁₀ 10Th.₁₀ 1000₁₀ 100₁₀ 10₁₀ 1₁₀

The Binary, Base₂, system has two discrete numeric values of 0 and 1₂, equivalent to 0 and 1₁₀.

Column values are shown for an 8-bit computer binary word, for a 16-bit word the MSB column would be 2¹⁵ (32,768₁₀).

Column Name (MSB)128s 64s 32s 16s 8s 4s 2s 1s (LSB)

Base₂ Column Value 2⁷ 2⁶ 2⁵ 2⁴ 2³ 2² 2¹ 2⁰

Decimal Column Value 128₁₀ 64₁₀ 32₁₀ 16₁₀ 8₁₀ 4₁₀ 2₁₀ 1₁₀

The Octal, Base₈, system has eight discrete numeric values of 0, 1₈, 2₈, 3₈, 4₈, 5₈, 6₈, and 7₈, equivalent to 0, 1₁₀, 2₁₀, 3₁₀, 4₁₀, 5₁₀, 6₁₀, and 7₁₀.

Column Name 32768s 4096s 512s 64s 8s 1s (Units)

Base₈ Column Value 8⁵ 8⁴ 8³ 8² 8¹ 8⁰

Decimal Column Value 32768₁₀ 4096₁₀ 512₁₀ 64₁₀ 8₁₀ 1₁₀

The Hexadecimal, Base₁₆, system has sixteen discrete alpha-numeric values of 0, 1₁₆, 2₁₆, 3₁₆, 4₁₆, 5₁₆, 6₁₆, 7₁₆, 8₁₆, 9₁₆, A₁₆, B₁₆, C₁₆, D₁₆, E₁₆, and F₁₆, equivalent to 0, 1₁₀, 2₁₀, 3₁₀, 4₁₀, 5₁₀, 6₁₀, 7₁₀, 8₁₀, 9₁₀, 10₁₀, 11₁₀, 12₁₀, 13₁₀, 14₁₀, and 15₁₀.

Column Name 65536s 4096s 256s 16s 1s (Units)

Base₁₆ Column Value 16⁴ 16³ 16² 16¹ 16⁰

Decimal Column Value 65536₁₀ 4096₁₀ 256₁₀ 16₁₀ 1₁₀

Example Convert 458₁₀ to Binary Base₂

Divide the number by 2 continuously until the value is 0.

2 ) 458 Remainder(R)

2 ) 229 (R)0

2 ) 114 (R)1

2 ) 057 (R)0

2 ) 28 (R)1

2 ) 14 (R)0

2 ) 07 (R)0

2 ) 3 (R)1

2 ) 1 (R)1

0 (R)1

Then read the binary value from the bottom (MSB) to the top (LSB) of the remainder column.

So 458₁₀ is 111001010₂

Example Convert 916₁₀ to Octal₈

Divide the number by 8 continuously until the value is 0.

8 ) 916 Remainder(R)

8 ) 114 (R)4

8 ) 14 (R)2

8 ) 1 (R)6

0 (R)1

Then read the octal value from the bottom to the top of the remainder column.

So 916₁₀ is 1624₈

Example Convert 1832₁₀ to Hexadecimal₁₆

Divide the number by 16 continuously until the value is 0.

16 ) 1832 Remainder(R)

16 ) 114 (R)8

16 ) 7 (R)2

0 (R)7

Then read the hexadecimal value from the bottom to the top of the remainder column.

So 1832₁₀ is 728₁₆

Converting Decimal Base₁₀ (458₁₀) to Binary Base₂

Converting Decimal Base₁₀ (916₁₀) to Octal Base₈

Converting Decimal Base₁₀ (1832₁₀) to Hexadecimal Base₁₆

Write the Base_n columns from the right-hand column (1s column or Binary LSB) moving left, adding more, until the Column Base₁₀ Value is greater than the decimal value to be converted (maximum required column or Binary MSB).

Write 0 in this final, maximum, column (discarded later),

Binary Base₂ –write 1 in the next column.

Octal Base₈ & Hexadecimal Base₁₆ – calculate the next column numerical value by dividing the decimal starting value by the column Base₁₀ value and write the integer obtained as the column numeric value.

Base₂

2⁹ 2⁸ 2⁷ 2⁶ 2⁵ 2⁴ 2³ 2² 2¹ 2⁰

512₁₀ 256₁₀ 128₁₀ 64₁₀ 32₁₀ 16₁₀ 8₁₀ 4₁₀ 2₁₀ 1₁₀

0 1

Base₈

8⁴ 8³ 8² 8¹ 8⁰

4096₁₀ 512₁₀ 64₁₀ 8₁₀ 1₁₀

0 1

Base₁₆

16³ 16² 16¹ 16⁰

4096₁₀ 256₁₀ 16₁₀ 1₁₀

0 7

Base₂ Subtract the decimal value of that column from the starting value

Base₂ 458₁₀ – 256₁₀ = Remainder 202₁₀

Base₈ & Base₁₆ Multiply the integer, the column numeric value, by the column Base₁₀ value and then subtract the result from the starting value

Base₈ 916₁₀ – 512₁₀ = Remainder 404₁₀

Base₁₆ 1832₁₀ – 1792₁₀ = Remainder 40₁₀

Move along all columns, writing 0 when the column Base₁₀ value is greater than (>) the remainder.

When the column Base₁₀ value is less than (<) the remainder –

Base₂ Write 1 then subtract the column Base₁₀ decimal value from the current remainder...

Base₈ & Base₁₆ Calculate the required column numeric value by dividing the remainder value by the column Base₁₀ value and write the integer obtained, as the column numeric value, then multiply the integer by the column Base₁₀ value and subtract the result from the current remainder...

... to produce a new remainder value.

Base₂

128₁₀ < 202₁₀ hence 2⁷ column = 1; 202₁₀ - 128₁₀ = 74₁₀ (new remainder)

64₁₀ < 74₁₀ hence 2⁶ column = 1; 74₁₀ - 64₁₀ = 10₁₀ (new remainder)

And so on resulting in the remaining columns being 0, 0, 1, 0, 1, 0

So 458₁₀ is 111001010₂

Base₈

64₁₀ < 404₁₀ hence 404₁₀ ÷ 64₁₀ = 6; 64₁₀ x 6 = 384₁₀; 404₁₀ - 384₁₀ = 20₁₀ (new remainder)

8₁₀ < 20₁₀ hence 20₁₀ ÷ 8₁₀ = 2; 8₁₀ x 2 = 16₁₀; 20₁₀ - 16₁₀ = 4₁₀ (new remainder)

And so on, resulting in the remaining column value being 4.

So 916₁₀ is 1624₈

Base₁₆

16₁₀ < 40₁₀ hence 40₁₀ ÷ 16₁₀ = 2; 16₁₀ x 2 = 32₁₀; 40₁₀ - 32₁₀ = 8₁₀ (new remainder)

And so on, resulting in the remaining column value being 8.

So 1832₁₀ is 728₁₆

Convert Binary Base₂ (111001010₂) to Octal Base₈

Group the binary digits into groups of three beginning at the right-hand side

111 001 010

Then convert each group to Decimal Base₁₀, equivalent Base₈, values,

712₈ [Check 2+(1x8)+(7x64)= 2+8+ 448= 458]

Convert Binary Base₂ (111001010₂) to Hexadecimal Base₁₆

Group the binary digits into groups of four beginning at the right-hand side

1 1100 1010

Then convert to Decimal Base₁₀, equivalent Base₁₆, values,

1CA₁₆ [Check 10+(12x16)+(1x16)= 10+192+ 256= 458]

Convert Binary Base₂ (111001010₂) to Decimal Base₁₀

First group the columns and then convert them to either Octal or Hexadecimal (personal preference), as above, and then convert to Decimal.

Convert Octal Base₈ (712₈) to Binary Base₂

Write out the numbers in groups of three binary digits

712₈ = 111001010₂

Convert Octal Base₈ (712₈) to Hexadecimal Base₁₆

Write out the numbers in groups of four binary digits

Then convert these groups to Hexadecimal Base₁₆ values

712₈ = 1 1100 1010 = 1CA₁₆

Convert Octal Base₈ (712₈) to Decimal Base₁₀

Calculate each individual column Base₁₀ value and sum them

712₈ = (7x64₁₀) + (1x8₁₀) + 2₁₀ = 458₁₀

Convert Hexadecimal Base₁₆ (916₁₆) to Binary Base₂

Write out the numbers in groups of four binary digits

916₁₆ = 1001 0001 0110₂ (without spaces)

Convert Hexadecimal Base₁₆ (916₁₆) to Octal Base₈

Write out the numbers in groups of four binary digits

916₁₆ = 1001 0001 0110₂

Then group them in threes

= 100 100 010 110₂

Then convert these groups to Octal Base₈ values

= 4426₈

Convert Hexadecimal Base₁₆ (916₁₆) to Decimal Base₁₀

Calculate each individual column Base₁₀ value and sum them

916₁₆ = (9x256₁₀) + (1x16₁₀) + 6₁₀ = 4118₁₀