Programmable Logic Controllers. Su Chen Jonathon Lin

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      Figure 3.3: A hexadecimal number

      The focus of this section is to show the techniques of converting a numerical quantity from one number system to another. A decimal number can be converted to binary, octal, and hexadecimal. Converting a decimal number to its equivalent in another number system involves the process of dividing the decimal number by the base number of the system from which the conversion is made. The conversion to binary involves division by 2, conversion to octal involves division by 8, and conversion to hexadecimal involves division by 16. The generic procedure involves:

       a.Dividing the decimal number by the base number.

      Decimal number ÷ Base number = Quotient (Q₁) + Remainder (R₁)

      Quotient (Q₁) ÷ Base number = Quotient (Q₂) + Remainder (R₂)

      Quotient (Q₂) ÷ Base number = Quotient (Q₃) + Remainder (R₃)

      Quotient (Q_n-1) ÷ Base number = Quotient (Q_n) + Remainder (R_n)

      Repeat the process until the quotient of the division becomes 0.

       b.Arranging the remainders in proper sequence to express the number in the new number system.

      Use the following tips to express the number in the new number system:

      •The remainder (R₁) is placed in the least significant digit (LSD, right-most) position of the new base number.

      •The remainder (R₂) is placed in the second position from the right of the new base number.

      •The last remainder (R_n) is the most significant digit (MSD, left-most digit) of the new base number.

      The new base number is expressed as (R_n… R₃R₂R₁)_{base number}

      This procedure can be applied to converting decimal numbers to all three other number systems: binary, octal, and hexadecimal.

       3.6.1Converting Decimal to Binary

      Converting a decimal number to its binary equivalent involves repeatedly dividing the decimal number by 2 until a quotient of 0 is obtained. Each remainder obtained is a digit of the equivalent binary number, starting from right to left.

       Example 3.1: Converting decimal 125 to its binary equivalent

      The procedure of converting 125₁₀ to its binary equivalent is outlined below. The binary equivalent of 125₁₀ becomes 1111101₂.

       Example 3.2: Converting decimal 70 to its binary equivalent

      The procedure of converting 70₁₀ to the binary number is listed below. The binary equivalent of 70₁₀ becomes 1000110₂.

       3.6.2Converting Decimal to Octal

      Converting a decimal number to its octal equivalent involves repeatedly dividing the decimal number by 8 until a quotient of 0 is obtained. Each remainder obtained is a digit of the equivalent binary number, starting from right to left.

       Example 3.3: Converting decimal 253 to its octal equivalent

      The procedure of converting 253₁₀ to octal number is as follows:

      The octal equivalent of 253₁₀ is 375₈.

       3.6.3Converting Decimal to Hexadecimal

      Converting a decimal number to its hexadecimal equivalent involves repeatedly dividing the decimal number by 16 until a quotient of 0 is reached. Each remainder obtained is a digit of the equivalent hexadecimal number, starting from right to left.

       Example 3.4: Converting decimal 895 to its hexadecimal equivalent

      The procedure of converting 895₁₀ to hexadecimal number is as follows:

      The hexadecimal equivalent of 895₁₀ becomes 37F₁₆.

       3.6.4Converting Binary to Decimal

      The procedure of converting a binary number to its decimal equivalent starts by multiplying the place values to their corresponding digits and summing the results. The procedure of converting binary number 1101₂ to decimal follows. We can see that 1101₂ = 13₁₀.

       Example 3.5: Converting 110112 to its decimal equivalent

      The decimal equivalent of 11011₂ is found as follows:

      11011₂ = 1x2⁴ + 1x2³ + 0x2² + 1x2¹ + 1x2⁰ = 16 + 8 + 0 + 2 + 1 = 27₁₀

       3.6.5Converting Octal to Decimal

      The procedure of converting an octal number to its decimal equivalent starts with multiplying the place values (1, 8¹, 8²,…, 8ⁿ) by their corresponding digits and summing the results. An example of converting octal number 534₈ to its decimal equivalent follows.

       Example 3.6: Converting 6278 to its decimal equivalent

      The decimal equivalent of 627₈ is:

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