The expanded notation It is one in which a numerical figure is expressed as a sum in which the positional value of each digit that makes up the number is taken into account.

For example, when a number such as 2345 is written, each digit in it has a positional hierarchy. Reading from the extreme right digit to the left, the hierarchy or value grows.

In the number 2345, the digit 5 represents five units, the digit 4 represents four tens, the 3 corresponds to the third position from left to the right and therefore the 3 represents three hundreds, and finally the 2 represents two thousands. In other words, in developed or extended notation the number 2345 is written like this:

2345 = 2 thousands + 3 hundreds + 4 tens + 5 ones

But it can also be expressed as follows:

2345 = 2 x 1000 + 3 x 100 + 4 x 10 + 5 x 1.

Also the number 2345 can be written as the sum of powers of 10:

2345 = 2 x 10^3 + 3 x 10^2 + 4 x 10^1 + 5 x 10^0

Where the circumflex ^ means to raise to the indicated exponent. For example, 10^3 = 10 x 10 x 10 = 1000. Another way to write exponents is using a superscript:

2345 = 2 x 103 +3×102 +4×101 +5×100

[toc]

## Positional number system

The Arabic numerals system are the numbers that are used daily in the vast majority of continents and countries in the world. Arabic numerals are a base 10 system because ten symbols or graphemes are used to write any number. These ten symbols are:

0 1 2 3 4 5 6 7 8 9

With only one of these symbols you can express numbers between zero and nine. To express digits greater than nine, the positional system in base ten is used. The number 10 is one ten and zero ones. The number 11 is a ten and a unit. The number 123 (one hundred twenty-three) is one hundred, two tens and three units. Written in the form of powers of ten the number 123 will be:

1×10^2 + 2×10^1 + 3×10^0

Where:

10^2 = 10 x 10 = 100

10^1 = 10

10^0 = 1.

With this example it is clear that the position of the digit on the far right is position 0 and represents the number of units, that of the second digit from right to left is position 1 and represents the number of tens, the third digit (from right left) has position 2 and represents the hundreds.

### Fractional or Decimal Numbers

With the decimal positional system it is also possible to represent numbers or figures that are less than the unit or that are greater than the unit but not whole, that is, they have fractions of the unit.

To represent the fraction ½ in the Arabic decimal system, that is, half of the unit, write:

½ =0.5

To arrive at this expression in our base 10 system, the following operations have been implicitly performed:

1- The numerator and denominator are multiplied by 5 to have the equivalent fraction 5/10 = 1/2.

2- Dividing by 10 is equivalent to multiplying by the power in base ten with an exponent minus one (10^-1), that is, 5/10 = 5×10^-1.

3- The negative exponent indicates how many times the indicated digit is shifted or positioned to the right from the unit position, in our case it would be 0.5.

4- ½ = 0.5 in extended notation is written like this:

0.5 = 0x10^0 + 5×10^-1

Where 10^-1 = 0.1 is a tenth (the fraction corresponding to the unit divided into 10 equal parts).

In this way, the number 0.5 corresponds to five tenths, but the number 0.05 corresponds to 5 hundredths and 0.005 to 5 thousandths.

## Examples of expanded notation

### Example 1

Given the number 40201 in standard notation, convert it to expanded notation.

Solution:

4×10000 + 0x1000 + 2×100 + 0x10 + 1×1 = 40201

### Example 2

Write the fraction ¾ in extended notation.

Solution:

In this case you have three quarters of the unit.

3/4 = 15/20 = 75/100 = 0.75 = 7/10 + 5/100 =

7×10^-1 + 5×10^-2.

Put in words it would look like this:

The fraction ¾ corresponds to seven tenths plus five hundredths.

## Expanded notation exercises

### Exercise 1

Say in words the expanded expression of the number 40201 from example 1.

Solution:

The developed notation looks like this:

40201 = 4×10000 + 0x1000 + 2×100 + 0x10 + 1×1

Which in the language of words is said:

Four tens of thousands, plus zero thousands, plus two hundreds, plus zero tens, plus one unit.

### Exercise 2

Express the previous figure in words and break down the corresponding phrase in developed form.

Solution:

The number 40201 in words is expressed as follows:

forty thousand two hundred one

The previous sentence can be developed as:

40×1000 + 2×100 + 1

It can be said that the way of pronouncing the figures is a semi-developed way of expressing it.

### Exercise 3

Write the number 7/3 in expanded form.

Solution:

It is a figure expressed as an improper fraction, since since the numerator is greater than the denominator, the figure is greater than unity.

This improper fraction can be decomposed as the sum of the fractions 6/3 + 1/3. The first of the fractions results in the integer 2, while 1/3 = 0.333333, where the digit 3 repeats indefinitely. So the expanded decimal expression of the number 7/3 will always be an approximate expression:

7/3 = 2 + 1/3 ≃ 2 + 0.333 = 2 + 3×10^-1 + 3×10^-2 + 3×10^-3.

### Exercise 6

Write the number in standard notation and then in expanded form: Twenty-three billion two hundred fifty million five hundred twenty-six thousand three hundred twenty-five and three twenty-three thousandths.

Solution:

It should be remembered that a billion is the equivalent of a billion. The word billion it was accepted by the Royal Spanish Academy in 1995 at the request of the late Venezuelan President Rafael Caldera, a member of the Venezuelan Academy of Language. In that case, the number of the exercise in standard notation is written like this:

23,2501526,325,023

23 billion + 250 million + 526 thousand + 325 units + 23 thousandths.

23×10^9 + 250×10^6 + 526×10^3 + 325×10^0 + 23×10^-3

Finally, the figure is written in expanded notation:

2×10^10 + 3×10^9 +2×10^8 + 5×10^7 + 0x10^6 + 5×10^5 + 2×10^4 + 6×10^3 + 3×10^ 2 + 2×10^1 + 5×10^0 + 0x10^-1 + 2×10^-2 + 3×10^-3.

## References

Khan Academy. Place value charts. Retrieved from: es.khanacademy.org

Khan Academy. Write a number in expanded form (video). Retrieved from: es.khanacademy.org

Ifrah, Geoges (1998): Universal history of figures. Espasa Calpe SA

Wikipedia. Positional notation. Recovered from: en.wikipedia.com

Wikipedia. Billion. Recovered from: en.wikipedia.com