1.9 Logic Gates and Computer Math

• ⭐️ not in exam

📌 Table

• 2 inputs and 1 outcome
• 2 inputs: A, B
• 1 outcome: Y

✅ AND

• outcome 1 only when both input is 1 & 1
• symbol: semi circle

✅ OR

• outcome 1 only when any input is 1
• symbol: rounded, curved triangle

✅ XOR

Exclusive OR

• outcome 1 only when one input is 1
• symbol: rounded, curved triangle with a curved line
• When there is too much electricity, the whole electricity goes off
• does not like too much 1s
• 🛠️ Used in CRC code
• 🛠️ Used in salt

✅ NOT

gate that makes the opposite

• ↔️ only one input and one output
• symbol: triangle

✅ NAND

opposite of AND
NOT AND

• outcome is opposite of AND
• draw AND and make opposite
• symbol: AND with a small circle, like NOT

00 ➡️ 1
01 ➡️ 1
10 ➡️ 1

• following the three first outputs,
• NAND is a gate that is active when smth is OFF
• When I unplug my USB(off), the data is not lost, data is active
• even when unplugged, the information is active
• so even when USB is unplugged, 1, which is 5v is active even when it is not connected to any electricity
• 🛠️ Thanks to NAND, we can have USB and SSDs
• NAND gates produces electrocity even when it is not plugged to electricity

✅ NOR

opposite of OR

• draw OR and make opposite

• symbol: OR with a small circle, like NOT

• NOR gate produces electricity when it is unplugged
• when it is 0, 0 result is 1
• but when it is plugged, cuts off the electricity
• ⚠️ NOR gate will save information when it is off, disconnected
• but they will lose information when conected
• when you connect/plug in, you delete/destroy data
• Thus, not valid for USBs
• 🛠️ Perfect for hardware malware
• create malware with hardware, so if somebody plugs in, you destroy all the information in his computer

✅ Shift Gate

moves the bits to the right

if 00010000 = 16
with shift gate
result will be 00001000 = 8

• 🛠️ Gate used for dividing in two
• Many dividing mathematical operations are done by Shift Gate

✅ Design gates

• normally gates come in groups
• 4 ANDs
• 4 NANDs

• 4 NANDs with shift function

• Logic gates are inside the micro processor/micro chip
• 🛠️ For doing all the mathematical operations for the computer

📌 How the computer adds

• we need to add in binary

  computer adds using XOR and AND use XOR for 0+0, 0+1, 1+0 and use AND for carrying

0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 0, and carry 1

• Looks exactly like XOR
• Thus, add is done by XOR gate

• when you buy XOR gate, your computer is adding

• we carry 1 when there are two 1s
• Thus, 1 + 1 carrying is done by AND

01011111 + 00110011
computer adds bit by bit
starting from the right
 - 1+1 result will be 0, and carry 1
👉🏻 10010010

• we might carry 1 + 1 at the very left end
• if that happens, we add the 1 at the right and add again

• so it might look like 1 looks small, but this is how computer adds

• 🛠️ When I use google maps, play games I am adding/substracting
• going closer: substracting distance, go further: add distance

📌 How the computer substracts

• for substraction, computer uses NOT and all the circuits for adding
• for substraction, computer uses NOT and XOR and AND
• the computer needs 2 XORs, 1 AND and 1 NOT
• so substraction needs more gates
• due to this, in programming, we try to do more adding then substraction

• if possible, we try to do more adding

• 1️⃣ We keep the first number exactly the way it is
• 2️⃣ we make the second number negative, use NOT
• 3️⃣ then we add both

15 - 12
1️⃣ 15
2️⃣ 12 ➡️ -12
3️⃣ = 15 + (-12)

010111 - 101011
1️⃣ 010111
2️⃣ 010100
3️⃣ 010111 + 010100 = 101011
👉🏻 101011

• 🛠️ Micro programming would be done in binary
• so adding, substraction would be done in Micro programming, in binary
• prefer adding