Place value, which you already use
Each column in a normal number is worth 10 times the column on its right. You do this in your head without noticing.
Take a number apart
Why does 9 turn into 10? There's no digit after 9. The ones column runs out, resets to 0, and carries 1 into the next column.
Binary columns
Binary has only 2 digits, so each column is worth double the one on its right.
Build the row yourself
Start at 1 on the right. Press the button to double it and add a new column on the left.
Count in binary
Same as normal counting, but you run out of digits after 1. Watch the carries.
128 · 64 · 32 · 16 · 8 · 4 · 2 · 1
Learn this row by heart. Almost every binary question uses it.
Binary → denary
For every column: if its bit is 1, count the column's value. If it's 0, skip it. Add up what you counted.
Tap the bits
The formula box below follows your bits. You can also type bits into it.
Why is 2⁰ = 1? Moving left doubles, so moving right halves. 2¹ = 2, and half of 2 is 1. That's 2⁰.
Denary → binary
Go through the columns from biggest to smallest. Ask one question each time: does this column's value fit into what's left? Yes = write 1 and take it away. No = write 0.
One question at a time
Hexadecimal, and why it exists
Binary is hard for people to read: 1001110011110101. Hex is a shorthand for it.
Why 16? 4 bits make 2⁴ = 16 patterns. So one hex digit stands for exactly 4 bits, and one byte (8 bits) is exactly 2 hex digits. Letters stand in for 10 to 15: A = 10, B = 11, C = 12, D = 13, E = 14, F = 15.
Split a byte into two hex digits
Blue box = the left digit. Orange box = the right digit.
Adding in binary
Add right to left, one column at a time, like normal. The twist: 2 is written 10 in binary, and 3 is written 11.
Add column by column
Overflow
A byte only holds 0 to 255. If an answer needs a 9th column, there isn't one, so that carry is lost.
A byte is like a clock
A clock goes round after 12. A byte goes round after 255, back to 0.
Binary shifts
Slide every bit one column to the left and the number doubles. Slide right and it halves. Like moving digits in denary: 35 → 350 multiplies by 10.
Slide the bits
Negative numbers (A-level bonus)
In two's complement, the leftmost column is worth −128 instead of +128. Everything else stays the same.
Same bits, two meanings
To write −5: write +5 (00000101), flip every bit (11111010), add 1 → 11111011.
Common mix-ups
The mistakes nearly everyone makes the first time. Read the left, then the right.
Binary 10 means ten.
Binary 10 is one 2 and no 1s = 2. Read it "one-zero", not "ten".
The leftmost bit is the 1s column.
The rightmost bit is the 1s column, just like the ones column in denary.
Hex is a different kind of data.
Hex is only a shorter way to write binary for people. The computer still stores bits.
Hex F = 16.
F = 15. The 16 digits are 0 to 15, because 0 counts.
Shifting right always halves exactly.
Odd numbers lose their last 1: 7 → 3. It rounds down.
Check yourself
Endless practice questions. Pick the kinds you want. A wrong answer shows you the working, step by step.