Convert Decimal to Binary

Convert Decimal to Binary

The Decimal or "denary" counting system uses the Base of 10 numbering system where each digit in a number takes on one of ten possible values from 0 to 9, eg 213₁₀ (Two Hundred and Thirteen). In a decimal system each digit has a value ten times greater than its previous number and this decimal numbering system uses a set of symbols, b, together with a base, q, to determine the weight of each digit within a number. For example, the six in sixty has a lower weighting than the six in six hundred and in a binary numbering system we need some way of converting Decimal to Binary.

Any numbering system can be summarised by the following relationship:

In the decimal or denary system, the columns have values of units, tens, hundreds etc as we move from right to left and mathematically these values are written as 10⁰, 10¹, 10², 10³ etc. The decimal numbering system has a base of 10 or modulo-10 (sometimes called MOD-10) with the position of each digit in the decimal system indicating the magnitude or weight of the number. For example, 20 (twenty) is the same as saying 2 x 10¹ and 400 (four hundred) is the same as saying 4 x 10². Likewise, for fractional numbers the weight of the number is negative, 10^-1, 10^-2, 10^-3 etc.

The value of a decimal number is equal to the sum of the digits multiplied by their respective weights. For example:  N = 6163₁₀  (Six Thousand One Hundred and Sixty Three)  in a decimal format is equal to:

(6×10³) + (1×10²) + (6×10¹) + (3×10⁰) = 6163

Unlike the decimal numbering system which uses the base of 10, digital logic uses just two values or states, a logic level "1" or a logic level "0", so each "0" and "1" is considered to be a single digit in a Base of 2 or Binary numbering system. In the binary numbering system, each digit has a value twice that of the previous digit but can only have a value of either "1" or "0" therefore, q = "2" and the position of either a "0" or a "1" indicates its weight.

For example:

By adding together all the decimal number values from right to left at the positions that are represented by a "1" gives us:  (256) + (64) + (32) + (4) + (1) = 357₁₀ or three hundred and fifty seven in decimal.

Then, the binary array of digits 101100101₂ is equivalent to 357₁₀ in decimal or denary. As the decimal number is a weighted number, converting from decimal to binary will also produce a weighted binary number with the right-hand most bit being the Least Significant Bit or LSB, and the left-hand most bit being the Most Significant Bit or MSB. and we can represent these as.

Repeated Division-by-2 Method

Another method of converting Decimal to Binary number equivalents is to write down the decimal number and to continually divide by 2 (two) to give a result and a remainder of either a "1" or a "0" until the final result equals zero.

Example.  Convert the decimal number 294₁₀ into its binary number equivalent.

Then, the decimal to binary conversion gives the decimal number 294₁₀ equivalent of 100100110₂ in binary, reading from right to left.

Then the main characteristics of a Binary Numbering System is that each "digit" or "bit" has a value of either "1" or "0" with each digit having a weight or value double that of its previous bit starting from the lowest or least significant bit (LSB) and this is called the "sum-of-weights" method. So we can convert a decimal number to binary either by using the sum-of-weights method or by using the repeated division-by-2 method.

Binary Number Names & Prefixes

Binary numbers can be added together and subtracted just like decimal numbers with the result being combined into one of several size ranges depending upon the number of bits being used. The classification of individual bits into larger groups are generally referred to by the following more common names of:

Also, when converting from Decimal to Binary or even from Binary to Decimal, we need to be careful that we do not mix up the two sets of numbers. For example, if we write the digits 10 on the page it could mean the number "ten" if we assume it to be a decimal number, or it could equally be a "1" and a "0" together in binary, which is equal to the number two in decimal from above. One way to overcome this problem and to identify whether the digits or numbers being used are decimal or binary is to write a small number called a subscript after the last digit to show the base of the number system being used. If we were using a binary number we would add the subscript 2 so the number would be written as 10₂, and if it were a decimal number we would add the subscript 10 so the number would be written as 10₁₀.

Today, as micro-controller or microprocessor systems become increasingly larger, the individual binary digits (bits) are now grouped together into 8´s to form a single BYTE with most computer hardware such as hard drives and memory modules commonly indicate their size in Megabytes or even Gigabytes.

Decimal to Binary Summary

• A "BIT" is the abbreviated term derived from BInary digiT
• A Binary system has only two states, Logic "0" and Logic "1"
• A Decimal system uses 10 different digits, 0 to 9
• A Binary number is a weighted number who's weighted value increases from right to left
• The weight of a binary digit doubles from right to left
• A decimal number can be converted to a binary number by using the sum-of-weights method or the repeated division-by-2 method
• When we convert numbers from decimal to binary, subscripts are used to avoid errors

Converting Decimal to Binary numbers can be done in a number of different ways as shown above. When converting decimal numbers to binary numbers it is important to remember which is the least significant bit (LSB), and which is the most significant bit (MSB). In the next tutorial about Binary Logic we will look at converting binary numbers into Hexadecimal Numbers and vice versa and show that binary numbers can be represented by letters as well as numbers.