# What is the binary form of 15

## Binary numbers

- introduction
- Counting in the decimal system
- Counting in the dual system
- Counting from 0 to 15 in decimal and binary systems
- Another notation
- Use of the power notation for binary numbers
- The conversion from binary to decimal numbers
- The conversion from decimal to binary numbers
- What is a bit?
- What is a byte?
- Kilobytes, megabytes, gigabytes, terabytes, petabytes, exabytes, zettabytes, yottabytes
- Calculating with binary numbers: addition, subtraction, multiplication, division

### introduction

The **dual number system** - also **Dual system** or**Binary system** called - consists of 2 numbers, marked by 0 and 1. This number system is needed in computer science, since the states ON and OFF can be generated very easily with technical components. These numbers can be used according to our "normal" decimal system. You can add, subtract, multiply and divide them. Since they hardly differ from "normal" arithmetic, they are ideally suited to be used in IT.

### Counting in the decimal system

We start at 0 and then count 1, 2, 3, etc. to 9. Now we're running out of numbers! To be able to continue counting, you now start with a 1 and start again at 0. That results in 1 and 0, i.e. 10. It continues with 11, 12, 13 to 19. The counting works up to 99, from now on we add another number, i.e. 100.

### Counting in the dual system

Again, we start with 0 and then count 1. Unfortunately, we only have 2 numbers, so we run out of numbers here quickly. But now we do it exactly as in the decimal system and add a digit. So after 0 and 1 there are 10 and 11. Again, the digits are not enough! So one more: 100, 101, 110, 111, etc.

### Counting from 0 to 15 in decimal and binary systems

Decimal Dual 0 0 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### Another notation

You can also represent numbers based on their base. In the decimal system we have 10 numbers available, from 0 to 9. With 2 digits we can represent 10 * 10 = 100 numbers. 100 numbers? But 100 has three digits! This objection is true. However, since we start with the number 0, 0 is the 1st number, 1 is the 2nd number, ... 98 is the 99th number and 99 is the 100th number. With 3 digits we can represent 10 * 10 * 10 = 1000 numbers. Each digit corresponds to a power of 10. I will try to explain this fact using a simple example. We take the number 372 and write it down as a small calculation: 372 = 3 * 100 + 7 * 10 + 2 * 1. This can now be represented differently than: 3 * 10^{2} + 7*10^{1} + 2*10^{0}.

You can now represent all other numbers in this way:

6574 = 6*10^{3} + 5*10^{2} + 7*10^{1} + 4*10^{0}

12032 = 1*10^{4} + 2*10^{3} + 0*10^{2} + 3*10^{1} + 2*10^{0}

### Use of the power notation for binary numbers

If you use the power notation for binary numbers, you have to choose a different base. There are only 2 different digits, 0 and 1. So we take 2 as the base. The number 1011 is then written as 1 * 2^{3} + 0*2^{2} + 1*2^{1} + 1*2^{0}

### The conversion from binary to decimal numbers

The power notation can now be used to convert binary into decimal numbers. If we now do the small calculation in the normal way, we get the corresponding decimal value: 1011 = 1 * 2^{3}+ 0*2

^{2}+ 1*2

^{1}+ 1*2

^{0}= 1 * 8 + 0 * 4 + 1 * 2 + 1 * 1 = 11 The dual number 1011 corresponds to the decimal number 11. (Compare with the table above)

### The conversion from decimal to binary numbers

When converting the decimal numbers, we use the "division with remainder" from elementary school. We divide the number by 2 until the result is 0 and note the rest. The numbers 13 and 14 should serve as an example.13/2 = 6 remainder 1

6/2 = 3 remainder 0

3/2 = 1 remainder 1

1/2 = 0 remainder 1

The remainders lined up from bottom to top then result in the dual number 1101.

14/2 = 7 remainder 0

7/2 = 3 remainder 1

3/2 = 1 remainder 1

1/2 = 0 remainder 1

This then results in the binary number 1110.

### What is a bit?

A bit is the smallest storage unit in IT. Information can be stored in one bit. This information can have two states, namely ON or OFF, i.e. 1 or 0. However, since you can do relatively little with this information, bits have been combined into bytes.### What is a byte?

A byte is the combination of 8 bits. With 1 byte, i.e. 8 bits, you can represent 256 different states. Why 256 different states? The smallest number that can be represented with 8 bits is the decimal 0, in dual notation 00000000. The largest number that can be represented is the decimal 255, in dual notation 11111111. We check this using the power notation:

11111111

= 1*2^{7} + 1*2^{6} + 1*2^{5} + 1*2^{4} + 1*2^{3} + 1*2^{2} + 1*2^{1} + 1*2^{0}= 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1

= 255

Again, 1 to 255 are 255 numbers. Plus the 0 we are at 256 different numbers or 256 different states. The term byte originally comes from the fact that a processor could read a maximum of one "bite", i.e. 8 bits at a time, from the memory in one computing step. This term is actually out of date with the new processor architectures, but has become firmly established.

### Kilobytes, megabytes, gigabytes, terabytes, petabytes, exabytes, zettabytes, yottabytes

A kilobyte is roughly a thousand bytes. More precisely: 2^{10} Byte = 1,024 bytes

A megabyte is approximately one million bytes. More precisely: 2^{20} Byte = 1,048,576 bytes

A gigabyte is roughly one billion bytes. More precisely: 2^{30} Byte = 1,073,741,824 bytes

A terabyte is approximately one trillion bytes, more precisely 2^{40} Byte = 1,099,511,627,776 bytes

A petabyte is roughly one quadrillion bytes. Exactly 2^{50} Byte = 1,125,899,906,842,624 bytes

An exabyte is roughly one trillion bytes. Exactly 2^{60} Byte = 1,152,921,504,606,846,976 bytes

A zettabyte is approximately one trillion bytes. Exactly 2^{70} Byte = 1,180,591,620,717,411,303,424 bytes

A yottabyte is roughly one quardillion bytes. Exactly 2^{80} Byte = 1,208,925,819,614,629,174,706,176 bytes

### Calculating with binary numbers: addition, subtraction, multiplication, division

The four basic arithmetic operations are described on the following pages:

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