Converting Number Systems

Stive has a 1st Class Honors Electronics Engineering degree including a high level math module and also masters level modules, from the O.U.

Common Number Systems Refresher

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

Converting Decimal Base10 to Binary Base2, (the quicker way)

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

Converting Decimal Base10 to Octal Base8, (the quicker way)

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

Converting Decimal Base10 to Hexadecimal Base16, (the quicker way)

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

Longer Method of Converting, understanding the columns

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

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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

Converting Binary Base2 to Octal Base8, Hexadecimal Base16 and Decimal Base10

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.

Converting Octal Base8 to Binary Base2, Hexadecimal Base16 and Decimal Base10

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)

Converting Hexadecimal Base16 to Octal Base8 and Decimal Base10

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

© 2019 Stive Smyth

Comments

Stephen on June 28, 2020:

Great site.