Methodological Instructions

Theme: Conversion between number system

Objective: 10.2.1.1 Convert decimal numbers to binary numbers and vice versa

Assessment criteria

·                    Translates numbers from decimal to binary

·                    Translations of the number of one number system to another

Basic Level:

Multiplication table (7-9 grade)

Key words and phrases:

Numeral system, binary, hexadecimal, octal and decimal system alphabet and how translate between numeral systems and vice versa, Numeral system is a ____

Translate number from binary system to decimal system…

I.                   THEORY

The decimal (base ten) numeral system has ten possible values (0, 1,2,3,4,5,6,7,8, or 9) for each place-value. In contrast, the binary (base two) numeral system has two possible values represented as 0 or 1 for each place-value.^[1] Since the binary system is the internal language of electronic computers, serious computer programmers should understand how to convert from decimal to binary.

For this example, let's convert the decimal number 156₁₀ to binary. Write the decimal number as the dividend inside an upside-down "long division" symbol. Write the base of the destination system (in our case, "2" for binary) as the divisor outside the curve of the division symbol.

·                                 This method is much easier to understand when visualized on paper, and is much easier for beginners, as it relies only on division by two.

·                                 To avoid confusion before and after conversion, write the number of the base system that you are working with as a subscript of each number. In this case, the decimal number will have a subscript of 10 and the binary equivalent will have a subscript of 2.

Divide. Write the integer answer (quotient) under the long division symbol, and write the remainder (0 or 1) to the right of the dividend.^[2]

·                                 Since we are dividing by 2, when the dividend is even the binary remainder will be 0, and when the dividend is odd the binary remainder will be 1.

Continue to divide until you reach 0. Continue downwards, dividing each new quotient by two and writing the remainders to the right of each dividend. Stop when the quotient is 0.

Write out the new, binary number. Starting with the bottom remainder, read the sequence of remainders upwards to the top. For this example, you should have 10011100. This is the binary equivalent of the decimal number 156. Or, written with base subscripts: 156₁₀ = 10011100₂

·                                 This method can be modified to convert from decimal to any base. The divisor is 2 because the desired destination is base 2 (binary). If the desired destination is a different base, replace the 2 in the method with the desired base. For example, if the desired destination is base 9, replace the 2 with 9. The final result will then be in the desired base.

Start by making a chart. List the powers of two in a "base 2 table" from right to left. Start at 2⁰, evaluating it as "1". Increment the exponent by one for each power. Make the list up until you've reached a number very near the decimal system number you're starting with. For this example, let's convert the decimal number 156₁₀ to binary.

Look for the greatest power of 2. Choose the biggest number that will fit into the number you are converting. 128 is the greatest power of two that will fit into 156, so write a 1 beneath this box in your chart for the leftmost binary digit. Then, subtract 128 from your initial number. You now have 28.

Move to the next lower power of two. Using your new number (28), move down the chart marking how many times each power of 2 can fit into your dividend. 64 does not go into 28, so write a 0 beneath that box for the next binary digit to the right. Continue until you reach a number that can go into 28.

Subtract each successive number that can fit, and mark it with a 1. 16 can fit into 28, so you will write a 1 beneath its box and subtract 16 from 28. You now have 12. 8 does go into 12, so write a 1 beneath 8's box and subtract it from 12. You now have 4.

Continue until you reach the end of your chart. Remember to mark a 1 beneath each number that does go into your new number, and a 0 beneath those that don't.

Write out the binary answer. The number will be exactly the same from left to right as the 1's and 0's beneath your chart. You should have 10011100. This is the binary equivalent of the decimal number 156. Or, written with base subscripts: 156₁₀ = 10011100₂.

·                                 Repetition of this method will result in memorization of the powers of two, which will allow you to skip Step 1.

Converting decimal to hexadecimal

·         There are several ways to convert decimal (sometimes called denary, or base 10) numbers into hexadecimal(base 16).

·         The video below gives a step by step guide on how to convert decimal numbers into binary code using the ‘divide by 16’ method.

Here’s how it works:

1. Divide the decimal number by 16.
2. If the number will not divide equally by 16, then round down the answer to the nearest whole number (integer).
3. Keep a note of the remainder, it should be between 0 and 15.
4. Keep repeating the above steps, dividing each answer by 16, until you reach zero.
5. Write out all the remainders, from bottom to top.
6. Finally, convert any remainders bigger than 9 into hexadecimal letters. This is your solution.

Decimal to Octal

Here is an example of using repeated division to convert 1792 decimal to octal:

Decimal to Hexadecimal

Here is an example of using repeated division to convert 1792 decimal to hexadecimal:

The only addition to the algorithm when converting from decimal to hexadecimal is that a table must be used to obtain the hexadecimal digit if the remainder is greater than decimal 9.

The addition of letters can make for funny hexadecimal values. For example, 48879 decimal converted to hex is:

П. TESTS AND ASSIGNMENTS FOR SELF-ASSESSMENT

1. Convert numbers from decimal to binary.

1. 17₁₀=?₂                                                                                                                             
2. 851₁₀=?₂

2. Convert numbers from binary to decimal.

1. 1111₂=?₁₀                                                                                                                          
2. 101110101₂=?₁₀                                                              

3. Convert real numbers from decimal to binary

a.       57,625                                                                           

_B.       2)11101.1011                                                                                           

VISUAL AIDS AND MATERIALS.

1.             Slides

2.             Introducing binary https://www.bbc.com/bitesize/guides/zwsbwmn/revision/7

3.             Different number systems http://www.anastasi-shherbakova.narod.ru/p7aa1.html

4.             Converting Denary to Binary  http://www.zaurtl.ru/UkVT/UKVT6.html

5.             Converting Binary to n-based system  https://www.youtube.com/watch?v=cPOgHCqecxY

6.             Abilities of the Calculator program: http://www.compgramotnost.ru/windows-7/kalkulyator-windows-7

7.             Converting Denary to Binary https://www.youtube.com/watch?v=70lM1qAD5u4

8.             Denary to Binary: Division https://www.sqa.org.uk/e-learning/CompArch01CD/page_10.htm

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