# How can I express 2 3 5

### Expand and shorten

You already know that fractions can have different names but still have the same value.

Is there definitely a connection between the various fractions? Sure, let's go:

### Shorten

If you look at the pictures of the two fractions, you will notice that there are no dividing lines in the picture on the right. It is a coarsening of the classification.

If you take a closer look at the fractions $$ 2/8 $$ and $$ 1/4 $$, you will notice that the numerator and denominator of the first fraction have been divided by 2.

$$2/8=(2:2)/(8:2)=1/4$$

This procedure is called **Shorten**.

A notation for shortening looks like this:

It means: $$ 2/8 $$ is shortened to $$ 2 $$. The two is under the equal sign. It means that you divide the numerator and denominator of the first fraction by two.

Trimming a fraction does not change the value of the fraction.

### The cut number

You can abbreviate with any number if you can divide up a whole number in the numerator and denominator.

You cannot shorten this task by 5, because then, unusual, you get:

Mathematically that's correct, but you're not talking about truncating when you get a decimal number.

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### Expand

Widening is the reverse of shortening. You refine a fraction.

You multiply the numerator and denominator by the same number.

Written down in detail it looks like this:

$$1/4 = (1*2)/(4*2) = 2/8$$

A notation for expanding looks like this:

$$ 1/4 stackrel (2) = 2/8 $$

The two on the equal sign means that you multiply the numerator and denominator by two.

When expanding, the value of the fraction does not change.

### Expansion numbers

You can expand a fraction with any natural number greater than 1.

$$ 2/3 $$ $$ stackrel (3) = $$ $$ 6/9 $$

### Shorten with rectangles

You can also represent the shortening and widening of rectangles.

**Example: shorten with 2**

Mathematically:

$$6/20 = (6:2) / (20:2) = 3/10$$

Or short:

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### Expand on routes

You can also represent the shortening and widening on routes.

**Example: Expand with 3**

$$ 1/3 stackrel (3) rarr 3/9 $$

Mathematically:

$$1/3 = (1*3) / (3*3) = 3/9$$

Or short:

$$ 1/3 stackrel (3) = 3/9 $$

### Shorten to the basic representation

You can truncate a fraction several times if the numbers allow.

If a fraction cannot be shortened further, this fraction is called the “basic representation” of the fraction.

If you can see the largest possible number with which you could cut right away, you can cut it in one step.

**Example:**

The divisibility rules will help you find the highest reduction number.

**Divisibility rules**

A number is divisible by 2 if its last digit is divisible by 2. (End point 0, 2, 4, 6, 8)

A number is divisible by 5 if its last digit is 0 or 5.

A number is divisible by 10 if its last digit is 0.

A number is divisible by 3 if its checksum (all digits +) is divisible by 3.

A number is divisible by 9 if its checksum (all digits +) is divisible by 9.

### Special cases 1 and 0

Expanding with the number 1 does not result in a new fraction. That's why you're not doing this. Shortening with 1 also leads to the same fraction. That too is superfluous, but possible.

Expanding with 0 is nonsense. If you multiply a number by 0, the result is 0. Shortening with 0 is not mathematically correct. Dividing by 0 produces no result.

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### Prime factorization and truncation

If you want to be very professional, you use the prime factorization of the numerator and denominator.

You write out the numerator and denominator in prime factors. All prime factors found in both the numerator and the denominator are truncation numbers. If you multiply them all together, you get the largest possible reduction number.

**Prime factor**

You can prime any number and express it as a multiplication problem. Example:

$$12 = 2* 2*3$$

$$ 2 $$ and $$ 3 $$ are prime numbers.

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