# Standard Form

In many fields, such as astronomy, extremely large numbers can be encountered. On the other hand, in fields such as nuclear physics, very small numbers are frequently dealt with. The problem with these numbers is that due to their magnitude, writing them in the mathematical form you are used to is extremely long, which takes up a large amount of physical space and is less comprehensible for the human eye.

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For example, the distance from the Earth to the Sun is approximately 150 million km. Written as a Number in metres, this gives us 150,000,000,000 m. This is already an extremely long number and we're only just scratching the surface; there exists many examples of much larger numbers in our universe.

How can this problem be solved? A way of writing numbers in a shortened form was invented to deal with this: standard form. This article will explain what standard form is and how to convert numbers to and from standard form.

## Standard form definition

Standard form is a way of writing numbers that allows for small or large numbers in a shortened form. Numbers in standard form are expressed as a multiple of a power of ten.

Numbers written in standard form are written in the form:

$A×{10}^{n}$

Where A is any Number greater than or equal to 1 and less than 10 and n is any integer (whole Number), negative or positive.

The exponent of 10 determines how big or small the number is, as larger positive exponents result in larger numbers:

${10}^{1}=10$

${10}^{2}=10×10=100$

${10}^{3}=10×10×10=1000$

${10}^{4}=10×10×10×10=10000$

Larger negative exponents result in smaller numbers:

${10}^{-1}=1/10=0.1$

${10}^{-2}=1/100=0.01$

${10}^{-3}=1/1000=0.001$

${10}^{-4}=1/10000=0.0001$

Is the following number written in standard form?

$12×{10}^{6}$

Solution:

The number is not written is standard form as A must be a number less than 10 and greater than or equal to 1. A is given as 12 which is greater than 10. This number in standard form would be $1.2×{10}^{7}$

## Standard form calculations

### Converting numbers into standard form

Numbers in standard form are written as a multiple of a power of 10. In the case of large numbers, the power of 10 will be large, meaning a positive exponent. For small numbers, the power of 10 will be extremely small (as multiplying a number by a decimal makes the number smaller), meaning a negative exponent.

In order to convert a number into standard form, follow these steps:

1. Move the decimal point until there is only one non-zero digit to the left of the decimal point. The number that has been formed is the value for A. For example, 5000 becomes 5.000, and we can remove the leading 0's giving us 5.
2. Count the number of times that the decimal point was moved. If the decimal point was moved to the left, the value for n in the formula will be positive. If the decimal point was moved to the right, the value for n in the formula will be negative. In the case of 5000, the decimal point was moved to the left 3 times, meaning n is equal to 3.
3. Write the number in the form $A×{10}^{n}$ using your results from step 1 and step 2.

### Converting numbers from standard form

In the case of converting numbers from standard form, we can simply multiply A by ${10}^{n}$, as standard form numbers are written as $A×{10}^{n}$.

For example, to convert $3.73×{10}^{4}$ from standard form, we multiply 3.73 by ${10}^{4}$. ${10}^{4}$ is the same as $10×10×10×10=10000$ , giving us $3.74×{10}^{4}=3.74×10000=37400$.

### Adding and subtracting numbers in standard form

The easiest way to add or subtract numbers which are written in standard form is to convert them into Real Numbers, perform the operation and then convert the result back into standard form. If you are permitted to use a calculator, these steps are not required as the calculator can perform the operation while displaying the result in standard form.

### Multiplying and dividing numbers in standard form

When multiplying and dividing numbers in standard form, the numbers can be kept in standard form, unlike with adding and subtracting. Follow these steps:

1. Perform the multiplication/division with the A of each number. This gives the A of the result.

2. If multiplying, add the exponents of 10 from each number together. If dividing, subtract the exponent of 10 from the 2nd number from the exponent of 10 from the 1st number. This is done because of the index laws.

3. You will now have a number in the form $A×{10}^{n}$. If A is 10 or more, or less than 1, you must convert the number back into a real number, and then back into standard form, so that the number is written in the correct standard form.

## Standard form examples

Convert the following number to standard form: 0.0086

Solution:

Firstly, we will move the decimal point until there is only one non-zero digit to the left of it. Doing this gives us 8.6, our value for A. We have moved the decimal point 3 places to the right, which means our value for n is -3. Writing the number in the form $A×{10}^{n}$ gives us:

$8.6×{10}^{-3}$

Convert the following number from standard form to an ordinary number: $4.42×{10}^{7}$

Solution:

${10}^{7}$ is the same as 10000000, as raising 10 to the power n gives a number with n zeros. To convert this number from standard form, we multiply 4.42 by 10000000, giving us $4.42×10000000$. If you are having problems with multiplying numbers by large powers of 10, simply multiply the number by 10 numerous times. In this case, we would multiply 4.42 by 10 seven times.

$4.42×{10}^{7}=44200000$

Calculate the following operation, giving your result in standard form: $8×{10}^{4}+6×{10}^{3}$

Solution:

Here we are being asked to add two numbers written in standard form together. Firstly, we convert the numbers from standard form into ordinary numbers:

$8×{10}^{4}=8×10000=80000$

$6×{10}^{3}=6×1000=6000$

Now we can proceed with the addition as normal using our numbers:

$80000+6000=86000$

Finally, we convert this number back into standard form. In this case, the decimal point is moved 4 places to the left, giving us a value of 8.6 for A and a value of 4 for n. Writing this in the form $A×{10}^{n}$ gives us our result:

$8.6×{10}^{4}$

Calculate the following operation, giving your result in standard form: $\left(1.2×{10}^{7}\right)÷\left(4×{10}^{5}\right)$

Solution:

In this question we must divide two numbers in standard form. Following our previously established steps, we shall begin by dividing the A value of each standard form number. $1.2÷4=0.3$. Next, we use index laws to perform the operation ${10}^{7}÷{10}^{5}$. This gives us ${10}^{7}÷{10}^{5}={10}^{7-5}={10}^{2}$.

Writing our number in the form $A×{10}^{n}$ gives us $0.3×{10}^{2}$. However, this is not yet written in standard form as A is less than 1! An easy way to fix this is by multiplying the value of A by 10, and subtracting 1 from the exponent. Or, we could also convert the number into an ordinary number and then convert this result into standard form:

$0.3×{10}^{2}=0.3×100=30$

Converting 30 into standard form:

Move the decimal point 1 to the left. This give us a value of 3 for A and a value of 1 for n. Writing this in the form $A×{10}^{n}$ gives us our answer:

$3×{10}^{1}$

## Standard Form (Ax10^n) - Key takeaways

• Standard form is a way of writing numbers that allows for small or large numbers in a shortened form. Numbers in standard form are expressed as a multiple of a power of ten.
• Numbers written in standard form are written in the form $A×{10}^{n}$, where A is any number greater than or equal to 1 and less than 10 and n is any integer (whole number), negative or positive.
• In order to convert a number into standard form, follow these steps:
1. Move the decimal point until there is only one non-zero digit to the left of the decimal point. The number that has been formed is the value for A.
2. Count the number of times that the decimal point was moved. If the decimal point was moved to the left, the number is positive. If the decimal point was moved to the right, the number is negative. This gives the value for n.
3. Write the number in the form $A×{10}^{n}$ using your results from step 1 and step 2.
• To convert a number $A×{10}^{n}$ from standard form to an ordinary number, multiply A by ${10}^{n}$.
• To add or subtract numbers which are written in standard form, convert them into Real Numbers, perform the operation and then convert the result back into standard form.
• To multiply or divide numbers in standard form:
1. Perform the multiplication/division with the A of each number. This gives the A of the result.
2. If multiplying, add the exponents of 10 from each number together. If dividing, subtract the exponent of 10 from the 2nd number from the exponent of 10 from the 1st number. This is done because of the index laws.
3. You will now have a number in the form $A×{10}^{n}$. If A is 10 or more, or less than 1, you must convert the number back into a real number, and then back into standard form, so that the number is written in the correct standard form.

#### Flashcards in Standard Form 4

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What is an example of standard form?

An example of a number written in standard form would be 5 x 103

What is standard form?

Standard form is a way of writing numbers  that allows for small or large numbers in a shortened form. Numbers in standard form are expressed as a multiple of a power of ten.

How do I write numbers in standard form?

In order to convert a number into standard form, follow these steps:

1. Move the decimal point until there is only one non-zero digit to the left of the decimal point. The number that has been formed is the value for A. For example, 5000 becomes 5.000, and we can remove the leading 0's giving us 5.
2. Count the number of times that the decimal point was moved. If the decimal point was moved to the left, the number is positive. If the decimal point was moved to the right, the number is negative. This gives the value for n. In the case of 5000, the decimal point was moved to the left 3 times, meaning n is equal to 3.
3. Write the number in the form Ax10^n using your results from step 1 and step 2.

How to transform this Standard Form (Ax10^n)?

In the case of converting numbers from standard form, we can simply multiply A by 10n, as standard form numbers are written as Ax10n.

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