JSS 3 First Term

Learning Objectives

• Explain place value in the decimal and binary systems.
• Convert numbers between decimal, binary and hexadecimal.
• Describe the octal system and what it is used for.
• Explain why computers use the binary system.

Starter Scenario

Amina's younger brother asks her, how does a computer count, it does not have fingers like we do. Amina thinks about her ten fingers and the decimal system she uses for counting, then wonders how a machine with only electrical signals can represent numbers at all. Computers use electric circuits that are either on or off, much like a light switch. An on state can represent 1 and an off state can represent 0. Using only these two digits, computers represent every number, letter, picture and sound. This is the binary system.

What is a Number System

A number system is a way of representing numbers using a specific base and a set of digits. The base of a number system tells you how many distinct digits it uses and how the value of each position increases. Every number system works on the same principle of place value. In the decimal system we use every day, each position from right to left multiplies by 10 because the decimal system has a base of 10.

The decimal system is base 10, meaning it uses ten digits from 0 to 9. Each position in a decimal number represents a power of 10. The positions from right to left are ones, which is 10 to the power 0, tens, which is 10 to the power 1, hundreds, which is 10 to the power 2, thousands, which is 10 to the power 3, and so on. Consider the number 2473. The digit 3 is in the units position with a place value of 1, giving 3 times 1 equals 3. The digit 7 is in the tens position with a place value of 10, giving 7 times 10 equals 70. The digit 4 is in the hundreds position with a place value of 100, giving 4 times 100 equals 400. The digit 2 is in the thousands position with a place value of 1000, giving 2 times 1000 equals 2000. Then 2473 equals 2000 plus 400 plus 70 plus 3. This same principle of place value applies to every number system, but the base changes.

The Binary System

The binary system is base 2, using only two digits, 0 and 1. Each position in a binary number represents a power of 2. The positions from right to left are ones, which is 2 to the power 0, twos, which is 2 to the power 1, fours, which is 2 to the power 2, eights, which is 2 to the power 3, sixteens, which is 2 to the power 4, and so on. Consider the binary number 1101. The digit 1 is in the ones position with a place value of 1, giving 1 times 1 equals 1. The digit 0 is in the twos position with a place value of 2, giving 0 times 2 equals 0. The digit 1 is in the fours position with a place value of 4, giving 1 times 4 equals 4. The digit 1 is in the eights position with a place value of 8, giving 1 times 8 equals 8. Then binary 1101 equals 8 plus 4 plus 0 plus 1, which is 13 in decimal.

Converting Decimal to Binary

To convert a decimal number to binary, repeatedly divide by 2 and record the remainders. The remainders, read from bottom to top, give the binary number. For example, to convert 13 to binary, first divide 13 by 2 to get 6 with remainder 1. Then divide 6 by 2 to get 3 with remainder 0. Then divide 3 by 2 to get 1 with remainder 1. Then divide 1 by 2 to get 0 with remainder 1. Reading the remainders from bottom to top gives 1101. So decimal 13 equals binary 1101. Learners can verify this by converting back, since 1101 equals 8 plus 4 plus 0 plus 1, which is 13.

The Octal System

The octal system is base 8, using digits from 0 to 7. Each position represents a power of 8. The positions are ones, which is 8 to the power 0, eights, which is 8 to the power 1, sixty-fours, which is 8 to the power 2, and so on. Octal is useful because every three binary digits can be grouped into one octal digit. This makes long binary numbers easier to read. For example, binary 000 equals octal 0, binary 001 equals octal 1, binary 010 equals octal 2, and so on up to binary 111 equals octal 7. To convert a binary number to octal, group the binary digits in threes from the right, then replace each group with its octal equivalent.

The Hexadecimal System

The hexadecimal system, often called hex, is base 16, using the digits 0 to 9 and the letters A, B, C, D, E and F. The letters stand for the decimal values 10, 11, 12, 13, 14 and 15. Every four binary digits map to exactly one hexadecimal digit, making binary strings much shorter and easier to read. For example, binary 0000 equals hex 0, binary 0001 equals hex 1, and so on through binary 1001 equals hex 9, binary 1010 equals hex A, binary 1011 equals hex B, binary 1100 equals hex C, binary 1101 equals hex D, binary 1110 equals hex E, and binary 1111 equals hex F.

Converting Between Systems

There are three main conversion directions learners should master. To convert binary to decimal, multiply each binary digit by its place value and add the results. For example, binary 1101 equals (1 times 8) plus (1 times 4) plus (0 times 2) plus (1 times 1), which equals 8 plus 4 plus 0 plus 1, which is 13. To convert decimal to binary, repeatedly divide by 2 and read remainders from bottom to top. For example, decimal 13 equals binary 1101. To convert binary to hexadecimal, group in fours from the right and replace each group. For example, binary 10110101 grouped as 1011 and 0101 equals B5 in hexadecimal.

Practical Work

1. Write the place values of the binary number 1101 and convert it to decimal.
2. Convert the decimal number 13 to binary by repeated division.
3. Convert the binary number 10110101 to hexadecimal.
4. Write the hexadecimal digits and their decimal values from 0 to F.

Scenario-Based Questions

1. A computer stores the number 1011 in binary. A pupil says this is the same as one thousand and eleven in decimal. Convert 1011 from binary to decimal to show the pupil the correct answer.
2. A school wants to label 16 network cables from 0 to 15 using hexadecimal numbers. Write out the 16 labels and explain why hexadecimal is shorter than using binary for the same labels.
3. A market trader counts her goods in tens and ones, for example 23 means 2 tens and 3 ones. Explain how this is like positional value in the decimal system and how binary uses the same idea with only 0 and 1.
4. A pupil converts the decimal number 13 to binary by dividing by 2 and gets 1101. Write out the step-by-step division showing the remainders to prove the answer is correct.
5. A radio station uses the number C4 in hexadecimal to label a frequency channel. Convert C4 to decimal and then to binary.
6. A teacher writes the number 77 on the board and asks the class to convert it to binary and to hexadecimal. Show both conversions.
7. Explain why computers use binary instead of decimal, linking your answer to the electrical state of a circuit.