**Convert between standard and scientific notation :**

A number N is in scientific notation when it is expressed as the product of a decimal number between 1 and 10 and some integral power of 10.

N = a x 10ⁿ, where 1 ≤ a < 10 and n is an integer.

The steps for converting a number to scientific notation are as follows:

**Step 1 :**

Move the decimal point so that there is only one non-zero digit to its left.

**Step 2 :**

Count the number of digits between the old and new decimal point. This gives n, the power of 10.

**Step 3 :**

If the decimal is shifted to the left, the exponent n is positive. If the decimal is shifted to the right, the exponent n is negative.

Let us see some examples based on the above concept.

**Example 1 :**

Write the following numbers in scientific notation.

205852

**Solution :**

Since the given number is integer, we have to consider that the decimal point is in the last place.

Here n = 5 because the decimal point is shifted five places to the left .

**Example 2 :**

Write the following numbers in scientific notation.

3449098.96

**Solution :**

Now we are going to move the decimal point so that there is only one non-zero digit to its left.

**Here n = 6 because the decimal point is shifted five places to the left .**

**Example 3 :**

Write the following numbers in scientific notation.

0.00008035

**Solution :**

Now we are going to move the decimal point to the right so that there is only one non-zero digit to its left.

Since we are moving the decimal point to the right side, we have to put negative sign in the power.

Often, numbers in scientific notation need to be written in decimal form. To convert

scientific notation to integers we have to follow these steps.

**Step 1 : **

Write the decimal number.

**Step 2 : **

Move the decimal point the number of places specified by the power of ten to the right if positive, to the left if negative. Add zeros if necessary.

**Step 3 : **

Rewrite the number in decimal form.

Let us see some examples problems based on the above concept.

**Example 4 :**

Write the following numbers in decimal form.

5.236 x 10⁵

**Solution :**

Since we have positive number as power, we have to move the decimal point to the right side.

**Example 5 :**

Write the following numbers in decimal form.

6.415 x 10⁻⁶

**Solution :**

Since we have negative number as power, we have to move the decimal point to the left side.

We move 6 digits to the left side, we have to write zero for the remaining digits.

**Example 6 :**

Write the following numbers in decimal form.

1.72 x 10⁹

**Solution :**

We have positive number as power, we have to move the decimal point to the right side.

1.72 x 10⁹ = 1720000000

Since we have to move 9 digits to the right, we have to put zero for the remaining digits.

**Example 7 :**

How do you write 119,000 in scientific notation?

**Solution :**

Since the given number is integer, we have to consider that the decimal point is in the last place.

Now we are going to move the decimal point to the left side, so we have to put positive sign in the power.

**119000 = 1.19 x 10**⁵

**Example 8 :**

How do you write 7.2 × 10² in standard form?

**Solution :**

Since we have positive number in the power, we have to move the decimal point to the right side.

7.2 × 10² = 720

**Example 9 :**

How do you write 2,300 in scientific notation?

**Solution :**

Since the given number is integer, we have to consider that the decimal point is in the last place.

Now we are going to move the decimal point to the left side, so we have to put positive sign in the power.

**2300 = 2.3 x 10**³

**Example 10 :**

How do you write 38,000 in scientific notation?

**Solution :**

Since the given number is integer, we have to consider that the decimal point is in the last place.

Now we are going to move the decimal point to the left side, so we have to put positive sign in the power.

**38000 = 3.8 x 10**⁴

After having gone through the stuff given above, we hope that the students would have understood "Convert between standard and scientific notation".

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