You probably know that
calculators and computers store decimal (0–9) numbers
as long strings of zeros and ones in a form called binary code.
Each number is stored using microscopic electronic switches called transistors.
It's easy to store binary numbers simply by switching transistors on
and off. Switching on a transistor stores a number one; switching it
off
stores a zero. So storing numbers is easy. But how can you add,
subtract, multiply, and divide using nothing but
electric currents?
Calculators and computers do this using clever
electronic circuits
called logic gates. Let's take a closer look
at what they are and how they work.
Photo: A gate can keep you out or let you
into a field. In the same way, a microscopic logic gate is a barrier in
an electronic circuit that can let electricity through or stop it
flowing altogether. Put lots of logic gates together and you make a
machine that's capable of basic mathematical "reasoning."
A logic gate might sound horribly complex, but it's
simply an electric circuit with two inputs and an output. It
receives two incoming electric currents, compares them, and sends on a
new, outgoing electric current depending on what it finds. A logic gate
is a bit like
a doorman or bouncer who is allowed to let people into a nightclub only
if they pass certain tests. There are quite a few different types of
logic gate, the most common of which are called AND, OR, NOT, XOR
(Exclusive
Or), NAND (NOT AND), and NOR (NOT OR). Let's look
at the three simpler ones, AND, OR, and NOT:
AND
Suppose you go to a nightclub where the doorman's job is to enforce
a simple rule: "Everyone in your group must wear a tie to come in". You
go along with
a friend one night. If you're both wearing ties, you'll get in. If only
one of you is wearing a tie, or if neither of you is, neither of you
will get in. An AND logic gate works the same way with two electrical
inputs. If both inputs are switched on (that is, carry a number 1), the
output will be 1 as well. Otherwise the output will be 0. In
electronics, we can represent an AND gate with this little symbol.
Three ways in which the gate can work are shown below.
OR
You're not wearing a tie, so you go to another club further down the
street. Here, the person on the door is enforcing a different
rule: "A group of people can come in if any one of them is a member".
If either you or your friend is a member, or if you both are members,
you can both come in. If neither of you is a member, you're
both left out in the cold. An OR logic gate works this way with two
electrical
inputs. If either input is switched on (that is, carries a number 1),
the output will be 1 as well. Otherwise the output will be 0. In
electronics, we represent an OR gate with a different symbol. Three
ways in which it can work are shown beneath:
NOT
So far, you've failed to get into either of the clubs. But
there's one last hope: you know a friend is having a party a few
streets away. The only trouble is, there's a really argumentative and
contrary person on the door. He talks to each person in turn as they
approach him. If you're nice and polite, he shouts abuse at you and
turns you away. But if you're rude to him, he likes that for some
reason and lets you in. In other words, he does exactly the opposite of
what you'd expect! In electronics, there's a logic gate that works in
the same, contrary way and it's called a NOT gate or inverter. Unlike AND
and OR gates, it has only one input and one output. The output is exactly the
opposite of the input, so if the input is a 0, the output is a 1 and
vice versa. Here's how we represent a NOT in electronics. Two ways it
can work are shown beneath.
The other three common logic gates are variations on these three.
XOR (Exclusive OR) is like an OR, but it switches off if both the
inputs are switched on. NAND is just like AND, only the end result is
swapped over (so where AND produces an output of 1, NAND produces an
output of 0). NOR is like OR with the end result swapped over in the
same way.
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Summarizing logic gates with truth tables
Computer chips don't really work like nightclub doormen: they just turn inputs into output—ones into zeros and zeros into ones. We can summarize how they work very quickly and simply by writing what's called a truth table. This shows the output you get (one or zero) for every possible input (each separate row of the table describes one of the possible sets of inputs and output). Here are the truth tables for the logic gates we met up above:
AND (Output only with all inputs)
Input 1
Input 2
Output
0
0
0
0
1
0
1
0
0
1
1
1
OR (Any input produces an output)
Input 1
Input 2
Output
0
0
0
0
1
1
1
0
1
1
1
1
NOT (Inverter)
Input
Output
0
1
1
0
XOR (Exclusive OR)
Input 1
Input 2
Output
0
0
0
0
1
1
1
0
1
1
1
0
NAND (AND with outputs reversed)
Input 1
Input 2
Output
0
0
1
0
1
1
1
0
1
1
1
0
Controlling a calculator display with logic gates
Calculators can do all the things they need to do using different
combinations of logic gates. It's logic gates that control how the
display works in a calculator and more logic gates that figure out the
results of calculations.
For example, each segment of a calculator's display is switched on and off by a series of
logic gates that are connected together. Consider just the bottom lower
right segment (shown in the picture, right, in blue). We need to turn
this segment on if we're showing the numbers 0
(binary 00), 1 (01), 3 (11), 4 (100), 5 (101), 6 (110), 7 (111), 8
(1000), and 9 (1001)—but not if we're showing the number 2 (10). We can
make the segment switch on and off correctly for the numbers 1–10 by
rigging up three OR gates and one NOT gate like this.
If we feed the patterns of binary numbers into the four inputs on
the left, the segment will turn on and off correctly for each one. For
example, feed in the number 7 as the four inputs 0111 and the gates
will trigger like this, correctly switching on the segment:
The other six segments of each digit are controlled by other
combinations of logic gates.
Calculating with logic gates
It's possible to do any calculation you can think of by combining a
different pattern of logic gates. The gates compare and combine two
series of binary numbers, corresponding to the decimal numbers in your
calculation, and produce a new series of binary numbers equal to the
result. Addition is carried out by circuits called half-adders
and full adders (which can carry on the
results of a previous addition); subtractors
work in a very similar way; multipliers
use a collection of AND gates to multiply two numbers with a series of
additions You can find more details about this in our article on
calculators.
Test your knowledge of logic gates
A good way to test your grasp of logic gates is to download one of the many apps that teach
you about them. Most of them are quite similar. After running through the
basics of how the different gates work, they present you with increasingly complex circuit patterns
and invite you either to work out what the final output will be or which gates to use to
achieve a certain output with a certain input. If you're new to circuit logic, I'd recommend trying one or two
just to make sure you've grasped the concepts; search for "logic gates" in your favorite app store.
Screenshot: Above) Circuit Scramble by Suborbital is an educational Android app that teaches you the basics of logic gates. In this very simple example, you have to figure out which of four inputs (bottom) to activate so that two OR gates and an AND gate produce an output at the top. Below) Logic Gates by Cyfrogen is a similar Android app. This time you're given the inputs (left), the output (right), and one of the gates, and you have to choose which other gate to use in the blank space (NAND or AND) to make the circuit work.
Who invented the idea?
“Boole was one of the most eminent of those who perceived that the symbols of operation could be separated from those of quantity and treated as distinct objects of calculation.”
Popular Science, October 1880.
This logical way of comparing numbers to make decisions that produce either a yes or no, 1 or 0, true or false is
called Boolean algebra after its discoverer, English mathematician George Boole (1815–1864),
who set out the idea in an 1854 book titled An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities.
His objective was to show how complex human reasoning could be represented in a logical, mathematical form.
Photo: George Boole, in an illustration from Popular Science Magazine, 1880
(with colors added).
Popular Science magazine paid tribute to him in its
October 1880 issue with these words:
“George Boole was a man of a very high order of genius, a profound and most original thinker of this century,
who will be known in future by his contributions to mathematical and logical science. Yet he can never be
widely known, for his work was so recondite that those who can properly appreciate it will always be
but very few.... An Investigation of the Laws of Thought... is Boole's greatest work, and is an
attempt to apply the symbols and operations of mathematics to logic and the laws of thinking.”
How logic gates add 2+3: How does a calculator add two binary numbers? This basic diagram shows you how four chips (made from AND, OR, and NOT gates) do the job. [Archived via the Wayback Machine.]
Adder (electronics): This more detailed (and less clear) article from Wikipedia explains how different logic gates can be used to add numbers in circuits called half adders, full adders, and more complex variations such as ripple-carry adders.
Books for younger readers
Computer Coding for Kids by Jon Woodcock et al. Dorling Kindersley, 2014. A fun overview of computer programming for young readers, including logic gates (covered on p186).
Electronic Circuits for the Evil Genius by Dave Cutcher. TAB/McGraw-Hill, 2011. Although not exclusively for young readers, the Evil Genius format is sure to appeal to teenagers. This book is a good hands-on introduction to digital logic and the projects include a variety of logic gate circuits.
Books for older readers
Electronics and computing
Digital Fundamentals by Thomas L. Floyd. Pearson, 2015. A clear introduction to digital circuits, including counters, logic gates, memory and storage, signal processing, and related topics.
Boolean Algebra and Its Applications by J. Eldon Whitesitt. Courier Dover, 2012. A reprinted book from 1961, this explains Boolean algebra with a mathematical approach based on sets, Venn diagrams, and so on. Unfortunately, it pre-dates modern computing so there's no real coverage of what we now call digital logic.
Celebrating Claude Shannon by G. Pascal Zachary. IEEE Spectrum, March 23, 2016. Shannon, one of the architects of our modern information age, drew inspiration from the work of George Boole—and helped Boole's work to find an appreciative modern audience.
Articles
These Optical Gates Offer Electronic Access by Charles Q. Choi. IEEE Spectrum. December 16, 2022. Light-based logic gates run a million times faster than electronic ones in experiments by researchers in Finland.
Boole Born, Boolean Logic Logically Follows by Dylan Tweney. Wired, November 2, 2010. A short introduction to the life and work of George Boole—and the impact it had on the world of computing.
The Logic Behind the Memristor by Willie D. Jones. IEEE Spectrum, April 13, 2010. Do memristors offer a credible alternative to silicon logic gates?
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