Computer ScienceCounting in Binary with Only Ones and Zeros
Byte the robot guide sits at a glowing circuit-board workbench, flipping rows of light switches labeled 1 and 0, watching numbers appear on a scoreboard overhead as each switch clicks on or off.
- Explain why computers use only two digits (0 and 1) instead of ten.
- Identify the place value of each bit position as a power of two.
- Convert a binary number up to 4 bits into its base-10 equivalent.
- Predict which bits must be switched on to represent a given number in binary.
Key terms
- Binary
- The base-2 number system using only the digits 0 and 1
- Bit
- A single binary digit that is either 0 or 1
- Place value
- The worth of a digit based on its column position
- Power of two
- A value like 1, 2, 4, or 8 found by doubling repeatedly
Why Two Digits Instead of Ten
Computers store everything in physical switches, and the most reliable switch has just two clear states: current flowing or not, voltage high or low. Two states map perfectly onto the digits 0 and 1, so the machine never has to tell apart ten slightly different voltage levels that noise could blur. Using only two symbols makes the electronics simpler, cheaper, and far less error-prone, which is exactly why every modern computer counts in base-2 rather than base-10.
Reading Place Values From Right to Left
Each binary column is worth twice the column to its right, so the values from the right are 1, 2, 4, 8, 16, and so on. To turn a binary number into base-10, write the place value above each digit, then add only the place values that sit above a 1. A 0 contributes nothing. This is the same idea as base-10, where columns are 1, 10, 100, except the multiplier between columns is two instead of ten.
Worked examples
Convert binary 1101 to base-10
- Write place values above the digits: 8, 4, 2, 1.
- Match each digit: 1->8, 1->4, 0->2, 1->1.
- Keep only the place values above a 1: 8, 4, and 1.
- Add them: 8 + 4 + 1 = 13.
Answer: 1101 equals 13 in base-10
Find which bits to switch on for 13
- Start with the largest place value not bigger than 13, which is 8; turn it on and subtract: 13 - 8 = 5.
- Next place value 4 fits in 5; turn it on and subtract: 5 - 4 = 1.
- Next place value 2 does not fit in 1; leave it off.
- Place value 1 fits in 1; turn it on and subtract: 1 - 1 = 0.
- Read the switches from 8 to 1: on, on, off, on.
Answer: 13 is binary 1101
Activity
Set each bit switch to ON (1) or OFF (0) to build the binary number that equals 13 in base-10. Choose a state for each column labeled 8, 4, 2, and 1.
Practice
Convert the binary number 0111 into its base-10 value.
Write the base-10 number 9 as a 4-bit binary number.
Common mistakes to avoid
- Binary 10 equals tenBinary 10 equals two, because the left column is worth two, not ten.
- Count the number of ones to get the valueYou must add the place values above each 1, not simply tally how many ones appear.
Check your understanding
What is the base-10 value of the binary number 1010?
A student says binary 1111 equals 4 because they count four 1s and treat that count as the answer. What is the correct value of 1111?
Which binary number correctly represents the base-10 value 6?
Recap
Computers use binary because switches have two reliable states. Each binary column is worth twice the one to its right, and you convert to base-10 by adding the place values that sit above a 1 while ignoring every 0.
Reflect
How might counting in only twos change the way you think about numbers?