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Binary number facts for kids

Kids Encyclopedia Facts
Binary 1-7
Numbers 0 to 7 shown in binary form.

The binary numeral system is a way to write numbers using only two digits: 0 and 1. These digits are used in computers as a series of "off" and "on" switches.

Most people use ten different digits — 0 to 9 — to write any number. For example:

  • 6
  • 52
  • 137
  • 9,826
  • 54,936

This is called the decimal number system or base ten system, which means that this number system has ten different digits to construct a number. It is a good system because people have 10 fingers to help them.

Computers do not use the decimal number system. This is because computers are built with electronic circuits, each part of which can be either on or off. Since there are only two options, they can only represent two different digits: 0 and 1. This is called the binary number system, or base two. ("Bi" means two.) All the numbers are constructed from the two digits 0 and 1. A digit in binary (that's a 0 or a 1) is also called a bit – short for binary digit.

Computers use this number system to add, subtract, multiply, divide, and do all their other math and data. They even save data in the form of bits.

A bit by itself can only mean zero or one, so to represent bigger numbers (and even represent letters), bits are grouped together into chunks. Eight bits make a byte, and computers use as many bytes as they need to store what we need them to. Modern computers have many billions of bytes of storage.

Why Do We Use Binary?

Height in binary numbers
Woman shows her height in binary numbers at the Copernicus Science Centre in Warsaw. (Credit: Alexander Baxevanis)

In normal math, we don't use binary. We were taught to use our normal number system. Binary is much easier to do math in than normal numbers because you only are using two number-symbols — 1 and 0 — instead of ten number-symbols — 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

With just two number-symbols, you can count quite high using things that just go "on" or "off," "yes" or "no". For example: How high can you count with your fingers? Most people would say that they could count to 10. If you count on your fingers with binary, you can count to 31 with one hand! With two hands, using binary, you can count up to 1023!

Computers use binary because they can only read and store an "on" or "off" charge. So, using 0 as "off" and 1 as "on," we can use numbers in electrical wiring. Think of it like this — if you had one color for every math symbol (0 to 9), you'd have ten colors. That's a lot of colors to memorize, but you have done it anyway. If you were limited to only black and white, you'd only have two colors. It would be so much easier to memorize, but you would need to make a new way of writing down numbers. Binary is just that — a new way to record and use numbers.

Binary Notation

In school, you were taught place value. Numbers have a ones place, a tens place, a hundreds place, a thousands place, a ten-thousands place, and so on. Each column, or place, is worth ten times more than the place to its right. If you haven't seen it before, place value looks like this:

Decimal column 10,000 1000 100 10 1
Digit 5 4 9 3 6
Value 5 × 10,000 4 × 1000 9 × 100 3 × 10 6 × 1

So the decimal number 54,936 is equal to 5×10000 + 4×1000 + 9×100 + 3×10 + 6×1.

Binary also has columns, but instead of multiplying by ten, they multiply by two each time:

Binary column 128 64 32 16 8 4 2 1
Bit 1 0 1 1 0 1 0 1
Value 1 × 128 0 × 128 1 × 32 1 × 16 0 × 8 1 × 4 0 × 2 1 × 1

So the binary number 10110101 = 1×128 + 1×32 + 1×16 + 1×4 + 1×1 = 181 in decimal.

This method lets us read binary numbers, but how do we write them? One way is to write a list of all the numbers starting from one and work upward. Just as adding 1 to 9 in decimal carries over to make 10 and 1 + 99 makes 100, in binary when you add one to one, carry a one over to the next place on the left. Follow along with this table to see how that works.

Base-10 Binary
1 1
2 10
3 11
4 100
5 101
6 110
7 111
8 1000
9 1001
10 1010
11 1011
12 1100
13 1101
14 1110
15 1111
16 10000

You'll notice that the values 1, 2, 4, 8, and 16 only need one 1 bit and some 0 bits, because there's a column with that value, and we just have to set the bit in that column to 1.

Base-10 version Binary
1 1
2 10
4 100
8 1000
16 10000

Have you noticed a pattern in writing binary numbers? Study the table for 1 to 16 again until you understand why in binary,

"1 + 1 = 10" and "1 + 100 = 101"

in your own way.

Since we use the decimal system, you may have lots of practice reading decimal but none yet reading binary. Reading binary may seem difficult and time-consuming at first, but with practice, it will become easier.

Translating to Base-10

The binary number for 52 is 110100. How do you read a binary number?

  1. First, we at the ones column. Since it has a 0 in it, we don't add anything to the total.
  2. Next, we look at the twos column. Nothing, so we move on to the next column.
  3. We have a 1 in the fours column, so we add 4 to the total (total is 4).
  4. Skipping the eights column since it has a 0, we have come to a 1 in the sixteens column. We add 16 to the total (total is 20).
  5. Last, we have a 1 in the thirty-twos column. We add this to our total (total is 52).

We're done! We now have the number 52 as our total. The basics of reading a base-2 number is to add each column's value to the total if there is a 1 in it. You don't have to multiply like you do in base-10 to get the total (like the 5 in the tens column from the above base-10 example). This can speed up your reading of base-2 numbers. Let's look at that in a table.

Binary digit Column Binary digit's value
0 1 0
0 2 0
1 4 4
0 8 0
1 16 16
1 32 32
Total 52

Finding a Mystery Number

Now let's look at another number. The binary number is 1011, but we don't know what it is. Let's go through the column-reading process to find out what the number is.

  1. The ones column has a 1 in it, so we add 1 x 1 to the total (total is 1).
  2. The twos column has a 1 in it, so we add 1 x 2 to the total (total is 3).
  3. The fours column has a 0 in it, so we add 0 x 4 to the total (total is still 3).
  4. The eights column has a 1 in it, so we add 1 x 8 to the total (total is 11).

There are no more columns, so the total is the answer. The answer is 11!

Storing Text

Computers store everything in binary, including text. To do this, every letter, every punctuation character, and, in fact, a very large number of the symbols people have ever used, has been given its own number in a system called Unicode.

For example, if your name is "George," then the computer can store that in binary just by storing the number for "G", then for "e", and so on. The most common symbols in American English, like letters without accents, can be stored with just one byte. Other symbols, like "£" and "¿", need more than one byte because they have been given a bigger number. A few examples:

  • G is stored as 71, which is "0100 0111" in binary.
  • e is stored as one hundred one, which is "0110 0101" in binary.

The whole word "George" looks like:

0100 0111 0110 0101 0110 1111 0111 0010 0110 0111 0110 0101

While this might look strange, see if you can find the rest of the letters in the word and what their decimal representation is!

History

Leibniz binary system 1697
The binary numeral system, in a manuscript by Gottfried Wilhelm Leibniz, 1697
Gottfried Wilhelm Leibniz, Bernhard Christoph Francke
The modern binary number system is credited to Gottfried Leibniz in 1679.

Binary was invented by many people, but the modern binary number system is credited to Gottfried Leibniz, a German mathematician, in 1679.

In 1817, John Leslie (a Scottish mathematician) suggested that primitive societies may have counted with objects (like pebbles) before they even had words to describe the total number of objects involved. Next, they would have discovered that this pile of objects could be reduced into two piles of equal measures (leaving either 0 objects or 1 object leftover). This remainder (odd = 1 or even = 0) would then be recorded and one of the piles removed, while the second pile was then further divided into two sub piles. If you record the remainder left over after the original pile has been divided in two and continue repeating this process, you will ultimately be left with just either 2 or 3 objects. If you record the remainder leftover (odd = 1 or even = 0) at the end of each reduction you will eventually be left with a tally record of 1's and 0's which will be the binary representation of your original pile of objects. So instead of representing your original pile of objects with a repeating number, or marks, or tokens (which for large numbers could be quite long), you have reduced your pile of objects into a more compact binary number. If you need to recover the original number of objects from this summarized binary number, it is easy enough to do: start with the first tally mark and then double it and add one if the next binary number contains a 1. Continue the process until the end of the binary number is reached. So, binary counting may be both the oldest and the most modern way of counting.

Binary has been used in nearly everything electronic; from calculators to supercomputers. Machine code is binary digits.

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Binary number Facts for Kids. Kiddle Encyclopedia.