--- title: "Why binary: logic gates" subtitle: "Data representation · Binary numbers · Lesson 7" --- # Why binary: logic gates **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** A computer chip holds billions of tiny switches and nothing else. How can switches add up numbers? :::teacher-note **In one sentence:** Students try every combination of two switches in a circuit to build AND and OR truth tables, use NOT, then add two bits with a half adder, to explain why digital systems are built on binary (AC9TDI8K04 node 3). **Students should be able to say:** I can give the output of AND, OR and NOT gates. I can explain why computers use binary. I can show how gates add two bits. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict :::keep-together **A battery, two switches and a lamp are wired in one loop, one switch after the other. Switch A is **on** and switch B is **off**. Does the lamp light?** ☐ No: the circuit is broken at B ☐ Dimly: half the switches are on ☐ Yes: one switch is enough ☐ Only if B was switched on first My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** No. With the switches in a line, the current has to pass through both, so B being off breaks the loop. The lamp lights only when A AND B are on. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Worked example: AND and OR :::keep-together ### Truth tables 1. $\text{AND}: 0\,0 \to 0,\quad 0\,1 \to 0,\quad 1\,0 \to 0,\quad 1\,1 \to 1$ — Switches in a line: the output is 1 only when both inputs are 1. 2. $\text{OR}: 0\,0 \to 0,\quad 0\,1 \to 1,\quad 1\,0 \to 1,\quad 1\,1 \to 1$ — Switches side by side: the output is 1 when at least one input is 1. 3. $\text{NOT}: 0 \to 1,\quad 1 \to 0$ — One input, flipped. 4. $\text{switch: 2 states} \to \text{bits}$ — Each switch is a bit: off 0, on 1. 5. $\text{gates} \to \text{adding, comparing, storing}$ — Wired together, gates do every calculation. So binary is the natural choice. Now read an OR gate. **Two switches side by side (an **OR** gate). A = 1 and B = 0. What is the output?** ☐ 1 ☐ ½ ☐ 0 ☐ It depends which switch was first ::: :::teacher-note **Answer:** 1 OR gives 1 when A or B is 1 (or both). A is 1, so the output is 1. ::: :::keep-together ### Find the mistake **Sam fills in the OR gate's truth table. Click the line where it first goes wrong.** ``` 1 Fill in the OR gate's output for each row. 2 A = 0, B = 0: neither is on, so 0. 3 A = 0, B = 1: one is on, so 1. 4 A = 1, B = 0: one is on, so 1. 5 A = 1, B = 1: OR means one or the other, not both, so 0. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 5 Line 5: OR gives 1 when at least one input is 1, so 1 OR 1 = 1. "One or the other but not both" is a different gate, XOR. ::: :::keep-together ### Why binary? **Why are computers built on binary, not on ten levels for the digits 0 to 9?** ☐ Computers can only count to 1 ☐ People invented binary because it is easy to read ☐ Switches have two states that are easy to tell apart, and gates can combine them to do any calculation ☐ Binary numbers are shorter ::: :::teacher-note **Answer:** Switches have two states that are easy to tell apart, and gates can combine them to do any calculation Two states are reliable even with a noisy signal, and switches wired as gates (AND, OR, NOT) can be combined to add, compare and store anything. ::: :::typical ## Vary: change the gate **OR with one on.** **OR with both on.** **NOT.** **AND's one row.** **Does B matter?.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Gates :::keep-together ### AND **What is **1 AND 0**?** ☐ 0 ☐ 1 ☐ 2 ☐ 10 ::: :::teacher-note **Answer:** 0 AND gives 1 only when both are 1. ::: :::keep-together ### NOT **What is **NOT 1**?** ☐ 0 ☐ −1 ☐ It needs two inputs ☐ 1 ::: :::teacher-note **Answer:** 0 NOT flips 1 to 0. ::: :::keep-together ### Alarm **A car alarm should sound if the door is opened **or** the window is broken. Which gate?** ☐ AND ☐ OR ☐ NOT ☐ None: alarms don't use gates ::: :::teacher-note **Answer:** OR Either input should set it off, so OR. ::: :::keep-together ### Machine guard **A machine should run only when the guard is closed **and** the start button is pressed. Which gate?** ☐ OR ☐ AND ☐ NOT ☐ XOR ::: :::teacher-note **Answer:** AND Both must be true: AND. ::: :::keep-together ### Review: binary **Review: what number is **110** in binary?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 6 110 = 4 + 2 = 6. ::: ## Make: add two bits Stretch: which inputs give the answer 01? Why are there two ways? Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Fill in the output: 1 AND 1, 0 OR 1, NOT 0. Why are computers built on binary?** {{lines-3}} ::: :::teacher-note **Answer:** 1, 1, 1. Switches have two reliable states, and gates combine them to do any calculation. :::