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SAT Subject Test Mathematics Level 2: Tutoring Solution24 chapters | 231 lessons

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Lesson Transcript

Instructor:
*Artem Cheprasov*

Would you like to learn how to write a gigantic or extra small number in a simple manner? This lesson provides very clear-cut rules on how to do so for any number you come across.

Do me a favor real quick. Tell me how large this number is and say what this number is out loud:

10000000000000000000000000000000000000000000.

Kind of ridiculous, isn't it? That number is so huge that virtually no one will know exactly what it is, even if there's a name for a number that big, like a million bazillion quintillion. That's not very convenient. This is why we write numbers in standard form. It's just much easier and faster.

**Standard form** can mean many different things, depending on which part of math we're actually dealing with. However, in this lesson's case, standard form is really another name for the scientific notation of a number. Think of the standard form/scientific notation as shorthand writing, but for math instead of note-taking. Scientific notation is really easy. In scientific notation we write a number in this format:

*A* x 10*b*

In this notation:

*A*is a number that's known as the coefficient. The coefficient must be greater than or equal to 1 but less than 10.- 'x' is the multiplication sign pronounced as 'times.'
- 10 is the base, and it must always be 10 in scientific notation.
*b*is a number that's the exponent, also known as the power of 10.

Thus, it would be read correctly as *A* x 10*b*.

For instance in this equation:

5.2 x 103

5.2 is the coefficient, 10 is the base, and 3 is the exponent. We read this number correctly as ''five point two times ten to the power of three.''

So how do we convert a number to standard form? Remember this one?

10000000000000000000000000000000000000000000

Preliminary step: if the number in question does not show a decimal point (the '.') already, put the decimal point at the very end of the number.

Step 1. Move the number's decimal point so that it's located right after the first non-zero digit in the number`:

1.0000000000000000000000000000000000000000000

Step 2. Count the number of places from this new decimal point to where the decimal point used to be. If you moved the decimal point to the left, the exponent is positive, if you moved it to the right, the exponent is negative. There are 43 leftward places from the new decimal point to the place where it was originally. This is your exponent: 43.

Step 3. Drop any zeroes that are located before the first non-zero digit and after the last non-zero digit to be left with your coefficient. All we're left with is 1 since this is our only non-zero number anywhere. This means our coefficient is 1.

Step 4. Combine steps 2 and 3 to get 1 x 1043, which is the same thing as 1043.

Boom! You're done.

Let's run through some more examples.

Convert the following into standard notation:

5675000000

The number has no decimal point, so we make it into this:

5675000000.

Next, we move the decimal point and place it right after the first digit:

5.675000000

Next, we count how many places we moved the decimal point. We moved it 9 leftward places. That's our exponent, 9.

Next, we drop all the zeros following the last non-zero digit in our number. What's the last non-zero digit? 5. So we drop all the zeros that follow it to get our coefficient of:

5.675

That means our new number is 5.675 x 109.

Let's try another example.

3154620007000000

It has no decimal point, so we make it into this:

3154620007000000.

Next, we move the decimal and place it right after the first non-zero digit:

3.154620007000000

Count how many places we moved the decimal: 15. That's our exponent. Drop all the zeroes following the last non-zero digit. Our last non-zero digit is 7, so we get our coefficient of:

3.154620007

Combine everything together to get 3.154620007 x 1015

12.5

This number already has a decimal point, so there's no need to do anything but move it to the place that's right after the first non-zero digit to get this:

1.25

How many places was the decimal point moved? 1 place. That's our exponent, 1. Do we need to drop any zeroes before our first non-zero digit or after our last non-zero digit? Nope, there are no zeroes before 1 and none after 5 in 1.25. So our coefficient is simply 1.25 Combine everything to get 1.25 x 101

1450.3506

Again, there's a decimal point there already, so just move it to get this:

1.4503506

How many places was it moved? 3, that's our exponent.

What's the last non-zero digit in this number? 6. Are there any zeros after it? Nope, there's nothing to drop so our coefficient is simply 1.4503506.

Combine to get 1.4503506 x 103 as the answer.

0.0000010053800000

This number already has a decimal point, so move the number's decimal point so that it's located right after the first non-zero digit in the number (which is 1):

0000001.0053800000

We moved the decimal point six places to the right, so the exponent is negative: -6.

Drop all the zeros that are located before the first non-zero digit and after the last non-zero digit to be left with your coefficient of this:

1.00538

Combine to get 1.00538 x 10-6.

The **standard form** of a number is a scientific notation written as:

*A* x 10*b*

Where *A* is the coefficient and *b* is the exponent.

If your number doesn't have a decimal point, add one to its very end. Move the decimal point to right after the first non-zero digit. Count how many places it was moved. That's your exponent. It's negative if it was moved to the right. Next, drop all zeroes before the first non-zero digit and after the last non-zero digit in the number. That's your coefficient.

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SAT Subject Test Mathematics Level 2: Tutoring Solution24 chapters | 231 lessons

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