Counting with just two digits: the binary system in motion

How to use this simulation

1. Look at the eight columns labelled 2⁷ down to 2⁰. Under each sits a box that is 1 (on, coloured) or 0 (off). Each box is a switch.
2. Press the small bit buttons under the controls, or drag the “Decimal number” slider. The boxes change, and the readings show the same number in decimal, binary, octal and hexadecimal.
3. Stay on “Add place values”. Play and the columns light up from the left; every column that is on contributes its place value, and the green line adds them.
4. Switch to “Divide by 2”. The number is halved again and again, and each remainder drops into a box on the right. Read the boxes from the bottom up to get the binary number.
5. Try “Octal” and “Hex”. The bits are cut into coloured groups of three or four from the right, and each group becomes one digit. Notice that no arithmetic on the whole number is needed.
6. Open “Binary addition” and set the second number. Columns are added from the right; a red 1 above a column is the carry. The line under the stage spells out each column.

Why computers count in twos

Inside a computer, everything is a switch. A transistor is either letting current through or it is not, a spot on a disc is either shiny or dull, a wire is either high voltage or low. Two states are easy to build, cheap to copy and hard to confuse. Ten states, one for each of our usual digits, would need ten different voltage levels and a circuit that could tell them apart reliably, even on a hot day with a bit of noise on the line.

So a computer writes numbers using only two symbols, 0 and 1, each called a bit (short for binary digit). It is the same counting you already know, just with a smaller alphabet. Once you see the pattern, converting between binary and decimal becomes a bit of routine arithmetic rather than a mystery.

Place value: the idea behind every number system

Take the decimal number 3,507. The 7 is worth seven ones, the 0 is worth no tens, the 5 is worth five hundreds and the 3 is worth three thousands. Each column is ten times the one on its right. The ten is the base, and it is why we call decimal base 10.

Binary uses exactly the same idea with a base of 2. Starting from the right, the columns are worth 1, 2, 4, 8, 16, 32, 64, 128 and so on, doubling each time. Because a digit can only be 0 or 1, a column is either counted once or not at all. There is no “three of this column”. That is the whole system.

Binary to decimal: add up the switched-on place values

Here is 214 written in binary: 11010110. Put it under the place-value columns and ask one question per column: is the switch on? If it is, take that column’s value, otherwise take zero. Then add.

The columns that are on are 128 + 64 + 16 + 4 + 2, which is 5 of the eight. Their sum is 214. Nothing else is needed. This is the “Add place values” view in the simulation.

Decimal to binary: halve it again and again

To go the other way, ask a different question: is the number odd or even? Divide by 2 and write down the remainder, which is 1 for odd and 0 for even. Then do the same with the quotient. Keep going until the quotient reaches 0. For 45 that takes 6 divisions.

The first remainder is the ones column (1) and the last one is the biggest column (1). So you read the remainders from the bottom to the top, which gives 101101. It is the reverse of the order in which you found them, and it is the single most common slip in this chapter.

The first sixteen numbers in four bases

Keep this table in your head and half of the chapter is free. Four bits cover exactly 0 to 15, which is why one hexadecimal digit stands for four bits. Past 9, hexadecimal runs out of digits and borrows the letters A to F, so A is ten, B is eleven and F is fifteen.

Octal: bits in groups of three

Binary numbers get long quickly. Octal is the shorthand that compresses them without any calculation, because eight is 2³ and so three bits make exactly one octal digit. Split the bits into threes starting from the right (pad the left with zeros if needed) and read each group on its own.

For 214 = 11010110, add a zero to the front and cut: 011 010 110. Those groups are worth the following digits, so 214 in octal is 326₈.

Hexadecimal: bits in groups of four

Hexadecimal is base 16, and sixteen is 2⁴, so every four bits make one hex digit. It is the favourite of programmers because one byte (eight bits) is always exactly two hex digits.

For 214, cut 11010110 into fours: 1101 0110. The left group is 13, which is written D, and the right group gives the other digit. Together that is D6₁₆.

Binary addition with carries

Binary addition has only four facts to remember: 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1 and 1 + 1 = 10 (that is, 0 and carry 1). It is the school column addition you already know, except that a column overflows as soon as it reaches 2 instead of 10.

Work from the right. Add the two bits and any carry in. If the total is 0 or 1, write it down. If it is 2, write 0 and carry 1 to the next column. If it is 3 (1 + 1 + 1 carry), write 1 and carry 1. Adding 101101 and 11011 (that is 45 + 27) needs 6 carries and gives 1001000, which is 72.

How many bits do I need?

With n bits you can write 2ⁿ different patterns, so the numbers 0 to 2ⁿ − 1. Add one more bit and the count doubles. This is why eight bits (a byte) give 256 values, and why 32-bit and 64-bit computers talk about memory in such huge numbers.

The bits needed for a number N is the smallest n with 2ⁿ greater than N. A quick way is to find the power of two just above it.

Try it in the simulation

Five quick experiments. Each takes under a minute and each one makes the rules stick.

• Switch on only the rightmost box: the number is 1. Switch on only the next: 2. Every single box is just a power of two.
• Turn all eight boxes on. You get the biggest one-byte number. Press the rightmost button to turn the 1 off and watch the number drop by exactly 1.
• Pick any number, run “Divide by 2”, then run “Add place values” on the same number. Two completely different routes, one answer.
• In “Hex”, find a number whose left group is 1111. Its hex digit is F, the highest single digit.
• In “Binary addition”, set both numbers to 127. Count the carries: the sum needs a ninth column, which is why this simulation stops at 127 for the second number.

Solved problems

Six common questions in the chapter. Every number below is computed with the same code as the simulation.

Mistakes to avoid

These cost marks again and again.

• Reading the remainders from top to bottom. They must be read bottom to top.
• Stopping the division at 1 instead of at 0. The last step 1 ÷ 2 gives quotient 0 and remainder 1, and that final 1 is your leftmost bit.
• Cutting octal or hex groups from the left. Always group from the right and pad the leftmost group with zeros.
• Writing 1 + 1 = 2 in binary. There is no digit 2. It is 10, which is 0 carry 1.
• Forgetting that hex letters are digits: A = 10, B = 11, C = 12, D = 13, E = 14, F = 15.
• Dropping the subscript base. Write 101₂ so no one reads it as one hundred and one.

Where binary is used

Every file on your phone is a long string of bits. A photo, a song and a text message are all stored as patterns of 0 and 1, and a byte (eight bits) is the usual unit. Kilobytes, megabytes and gigabytes are counted with powers of two.

In networking, addresses are binary numbers written in decimal for humans. In web design, a colour like #1E90FF is three hex bytes, one each for red, green and blue. In programming, a flag that is on or off is one bit. This is why the ICT chapters on computer systems, data representation, networks and web design keep coming back to this one idea.

Exam corner

For SSC, HSC and admission questions on number systems, remember these.

• Binary to decimal: write place values 1, 2, 4, 8 … from the right and add where the bit is 1.
• Decimal to binary: divide by 2 until you reach 0 and read the remainders upward.
• Binary to octal: groups of 3 from the right. Binary to hex: groups of 4 from the right.
• Octal or hex to decimal: multiply each digit by a power of 8 or 16 and add, or convert to binary first.
• Show the base as a subscript on every number and show every step. Method marks are awarded even when the last digit slips.

Frequently asked questions