Binary Math

What is binary math?

The binary number system is a base-2 number system. This means it only has two numbers: 0 and 1. The word binary simply means “two”, for example, this is a binary star system because it has two stars orbiting each other:

binary stars

The number system that we normally use is the decimal number system. It has 10 numbers: 0 through 9.

Why use binary math?

Binary numbers are important if you want to understand how computers work. Digital electronics (like computers) are dumb but they can easily understand being “ON” or “OFF”.

In terms of electricity, the binary number 1 is a high voltage while the binary number 0 is a low voltage, or ground – which is just a fancy way of saying “ON” and “OFF”. So at a fundamental level, you could think of computers as machines for flipping binary digits on and off, kind of like a lamp – it’s either on (light) or off (dark).

This binary “on/off” language is the computer’s equivalent of our human language:

binary-language

Everything you see or hear on the computer – words, pictures, videos and sounds – is made using only those zeros and ones. It’s hard to believe but it’s true!

Exercise: Counting In Binary

Here is a helpful table of the first 10 numbers to compare counting in decimal to counting in binary:

1 2 3 4 5 6 7 8 9 10
01 10 11 100 101 110 111 1000 1001 1010

Exercise: Cards

Make a couple sets of 5 cards like as shown below – just simple squares of paper with dots – and place them in the following order:

binary-cards

Q. Do you notice anything about the number of dots on each card?
A. Each card has twice as many dots as the card to the right.

Q. How many dots would there be on each card if we added two more cards to the left?
A. 32, 64

Q. In decimal math, if you add a zero to the end of a number, the number would be 10 times more. What happens in binary math?
A. The number doubles.

Now flip some of the cards face down, like this:

binary-cards-flipped

This is how we make binary numbers. The cards that are flipped down are zeros and the cards that are facing up are ones.

Q. If you count in binary, what number do you see if you flip the cards as shown above?
A. 9 (just count the dots!)

Q. How would you make the binary number “01001”?
A. Trick question! We already made a binary “01001” when we flipped the cards as shown above.

(Try flipping different cards to make different binary numbers. Hint: some numbers can be achieved using different combinations!)

Exercise: Faxes

Fax machines send and receive messages using the binary system too, by sending and receiving beeps. A high beep means a binary one and a low beep means a binary zero. Call any fax line right now and listen to it screech. It’s hard for our human ears to make out the high and the low beeps because they are being sent at high speeds but if you slowed the sound down, you would hear it.

Exercise: CDs and DVDs

CDs and DVDs use a binary system too, by having bits that reflect light and bits that do not (this is called optical storage). A shiny bit means a binary one and a matte bit means a binary zero. Go find a CD or a DVD and take a look at it under a light, turning it this way or that way. It’s hard for our human eyes to tell the shiny bits from the matte bits because they are so tiny but if you had a magnifying glass you would be able to see parts of the surface that reflect the light next to parts that don’t.

Bits and Bytes

Each of the cards we have used so far represents a “bit” on the computer – in fact “bit” is
just short for “binary digit“. In computer memory, the bits are made by a transistor that is switched on or off, or a capacitor that is charged or discharged.

One bit on its own is too small to be useful, so they are usually grouped together in groups of eight, which can represent numbers from 0 to 255. A group of eight bits is called a byte.

Take a break!

Breathe like this:

Breathe