# How is 2 a number

### Numbers in the tens system

Numbers you know so far are numbers in the **Decimal system**. This number system is also called the decimal system or decimal system.

There are 10 digits in the tens system:

0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

From the digits you write the numbers, like 3803.

Depending on where a digit is located, it has a certain value (place value). You can see that well in the status table.

1000s | 100s | 10s | 1er |
---|---|---|---|

On the right are the ones (1). Ten ones equals a ten (10), ten tens equals a hundred (100), ten hundreds equals a thousand (1000), and so on.

3803 is composed as follows:

3803 = 3 $$*$$ 1000 + 8 $$*$$ 100 + 0 $$*$$ 10 + 3 $$*$$ 1

A number in the system of ten is an ordered sequence of ten digits 0, 1, ..., 9. The place where the digit is used determines the value of the digit.

- The system of ten came to Europe from India with the Arabic language about 1000 years ago.
- For numbers in the tens system, you can also say “tens” or “decimal”.
- The order of the digits is important. 136 and 316 are made up of the same digits 1, 3 and 6, but they are different numbers.

### Computers calculate differently

But there are other computing systems as well. A computer does not calculate in the decimal system like you do. How then ?? !!!?

Computers only know two elements: 0 and 1. But with a sequence of zeros and ones you can **all numbers** write! This system is called **Twosystem**.

A number from the two-part system is 10111. So that you can see that it comes from the two-part system, you can write: (10111)_{2}

The place where the digit 0 or 1 is used determines the value of the digit (place value), as in the decimal system.

The place values are 1 and the powers of 2:

1, 2, 4, 8, 16, 32, 64, … .

The number 10111 in the place value table in the two-part system looks like this:

2, 4, 8 and so are tens.

The number 10111 is composed like this:

(10111)_{2} = 1 $$*$$ 16 + 0 $$*$$ 8 + 1 $$*$$ 4 + 1 $$*$$ 2 + 1 $$*$$ 1 = 23

Binary numbers consist of the digits 0 and 1. The place values are from right to left 1, 2, 4, 8, 16….

- Two numbers are also called
**Binary numbers**or**Binary numbers**. - Computers work with binary numbers. A digit in the binary system becomes
**bit**called. This is the abbreviation for Binary Digit. - The 1 stands for “current flows” and the 0 for “current does not flow”.

### From the dual representation to the tens representation

Here's how to convert a twos to a ten:

- Multiply the number by the corresponding place value.
- Add up all products from the number and the value.

**Examples**

- (110)
_{2}= 0·1+1·2+1·4 = 2+4 = (6)_{10} - (1011)
_{2}= 1·1+1·2+0·4+1·8 = 1+2+8 = (11)_{10} - (110101)
_{2}= 1·1+0·2+1·4+0·8+1·16+1·32 = (53)_{10}

The tens of a binary number is the sum of the place values with the number 1 in the binary number.

**Tens or twos?**

With the number sequence 101 you cannot tell whether it is the tens one hundred and one or a two number. To distinguish you write: **(101) _{2}** or

**(101)**.

_{10}*kapiert.de*can do more:

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### From the tens representation to the dual representation

How to convert a tens to a twos:

Determine if 1, 2, 4, 8, 16, 32, or other multiples of 2 are in the tens.

**Example:** Number in the tens system 27

Greatest multiple: 16 → 27 = **1** · 16 + 11

Next multiple: 8 → 27 = **1** · 16 + **1** · 8 + 3

Next multiple: 4 → 27 = **1** · 16 + **1** · 8 + **0** · 4 + 3

Next multiple: 2 → 27 = **1** · 16 + **1** · 8 + **0** · 4 + **1** · 2 + **1** · 1

The place value table:

Number in the decimal system: (27)_{10} = (11011)_{2}

If you add all the multiples of 2 contained in the binary number, you get the tens number.

- First find the largest possible multiple.
- If the multiple is included, the number 1 is in the two-digit representation.
- If the multiple is not included, the number 0 is in the two-digit representation.

### Second method

There is another way to convert tens to twos.

Divide the number of tens by 2 and write the rest down. Repeat with the result until you get to 0.

**example**

28: 2 = 14 remainder: 0

14: 2 = 7 remainder: 0

7: 2 = 3 remainder: 1

3: 2 = 1 remainder: 1

1: 2 = 0 remainder: 1

All remainders in reverse order are the digits of the dual representation.

**Result**: (28)_{10} = (11100)_{2}

This is how you can also convert a tens number into a binary number: Divide the tens number by 2 and write down the rest. Repeat with the result until you get to 0. Write all the remainders in reverse order and you have the binary number.

You can also use a number in tens.**example**

524: 10 = 52 remainder: 4

52: 10 = 5 remainder: 2

5: 10 = 0 remainder: 5

### Addition with binary numbers

And calculate with binary numbers? Sure, that works too!

You add binary numbers like decimal numbers. Dual numbers only consist of two digits 0 and 1, which you add up. 0 + 0 = 0 1 + 0 = 10 + 1 = 1 1 + 1 = 10

The last addition results in a carry of 1, which is added to the next digit addition.

**Example:**

1 1 0 1 0

+ 1 0 0_{1}1 1**Total: 1 0 1 1 0 1**

You can add numbers in the two-digit system in writing as in the ten-digit system with carryover.

You can add tens by adding the digits for the ones, tens, hundreds, and so on. If the sum is greater than 9, there is a carry that is added to the next digit addition.**example**3 6 8

+ 2_{1} 7_{1} 4**Total: 6 4 2**

*kapiert.de*can do more:

- interactive exercises

and tests - individual classwork trainer
- Learning manager

### Multiplication of binary numbers

And this is how you multiply:

You also multiply binary numbers digit by digit like decimal numbers.

0 · 0 = 01 · 0 = 00 · 1 = 01 · 1 = 1

The first factor is multiplied by the digits 0 or 1 of the second factor, one after the other, but either 0 or the first factor results.

**Example:**

1 1 0 1 · 1 0 1

1 1 0 1

0 0 0 0

1 1 0 1**Product: 1 0 0 0 0 0 1**

You can multiply numbers in the system of two in writing like in the system of ten.

You can multiply tens numbers in writing by individually multiplying the first factor with the digits of the second factor and adding the results.**example**3 6 8 · 2 3

7 3 6

1 1 0 4**Product: 8 4 6 4**

### Computers “calculate” in the binary system

Do you know **Bits** and **Bytes**?

All **Data** (Numbers, letters, characters) must be "translated" into a sequence of zeros and ones for processing and storage in the computer.

The size of the storage space is measured in **Bits** and **Bytes**.

**1 bit** is the smallest storage unit: A bit can be 0 or 1.

**1 B (byte) = 8 bit**

**1 KB (kilobytes)** = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 B = **1024 B**

**1 MB (megabytes)** = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 KB = **1024 KB**

**1 GB (gigabytes)** = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2MB = **1024 MB**

The memory size of USB sticks, memory cards, DVDs or hard drives is specified in bytes (usually in the order of gigabytes GB).

**kilo**, **Mega** and **Giga** are otherwise prefixes for thousands, millions and billions. **example**:

1 kg (kilograms) = 1000 g (grams)

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