Standard Form - Part 1

(How to write a number in standard form)

The idea behind standard form is to write both very large and very small numbers in a convenient form. Here are a couple of examples of numbers written in standard form:

• the distance from the Earth to the Sun is: $$149\ 600 \ 000 \ 000$$ m. In standard form this is written $$1.49 \times 10^{11}$$ m.
• The size of a Molecule of Oxygen is: $$0.000000000152$$ m. In standard form this is written $$1.52 \times 10^{-10}$$ m.
Writing numbers this way makes it much easier to see how big/small they are (as all we really have to focus on is how big/small the power of 10 is). Standard form also makes it much easier for us to compare two large (or small) numbers.

Tutorial 1: Writing Numbers in Standard Form

In this tutorial we learn how to write numbers in standard form. We do so by writing the three numbers:

• $$273\ 000\ 000$$
• $$0.0000273$$
• $$4\ 050$$
in standard form.

Standard Form

When we write a number in standard form, we must write it in yje following way: $a \times 10^k$ where the following two conditions must be met:

• $$1 \leq a < 10$$, the number $$a$$ must be greater than, or equal to, $$1$$ and must be less than $$10$$.
• $$k\in \mathbb{Z}$$, in other words the power on $$10$$ must be an integer: $$\mathbb{Z} = \left \{ \dots , -3, \ -2, \ -1, \ 0, \ 1, \ 2, \ 3, \ \dots \right \}$$

Exercise 1

Write each of the following numbers in standard form:

1. $$3,240,000$$
2. $$0.0000123$$
3. $$137,000$$
4. $$0.0074$$
5. $$607,000,000,000$$
1. $$0.000000103$$
2. $$9$$
3. $$0.67$$
4. $$43,000$$
5. $$0.00089$$
Note: this exercise can be downloaded as a worksheet to practice with:

Solution Without Working

We find the following results:

1. $$3,240,000 = 3.24 \times 10^6$$

2. $$0.0000123 = 1.23 \times 10^{-5}$$

3. $$137,000 = 1.37 \times 10^5$$

4. $$0.0074 = 7.4 \times 10^{-3}$$

5. $$607,000,000,000 = 6.07 \times 1O^{11}$$

6. $$0.000000103 = 1.03 \times 10^{-7}$$

7. $$9 = 9\times 10^0$$

8. $$0.67 = 6.7\times 10^{-1}$$

9. $$43,000 = 4.3\times 10^4$$

10. $$0.00089 = 8.9 \times 10^{-4}$$

Rounding & Standard Form

When writing numbers in standard form, we usually round numbers to two or three significant figures. For instance, consider the number: $2 \ 347 \ 128$ Without any rounding, writing this in standard form, leads to: $2.347128\times 10^6$ This notation isn't practical. Instead, when writing numbers in standard form, we usually round to 2 or 3 significant figures, so that $$2.347128\times 10^6$$ becomes:

• $$2.3\times 10^6$$ to two significant figures (2 sf)
• $$2.35\times 10^6$$ to three significant figures (3 sf)

Exercise 2

Writing all of your answers to three significant figures, write each of the following in standard form:

1. $$278,120,001$$
2. $$0.0004352617$$
3. $$31,906,321,003$$
4. $$0.000000789654$$
5. $$301,888$$
1. $$0.010345$$
2. $$978,909,456,001,745$$
3. $$0.0050216$$
4. $$8,927$$
5. $$0.435269$$
Note: this exercise can be downloaded as a worksheet to practice with:

Solution Without Working

We find the following results:

1. $$278,120,001 = 2.78 \times 10^6$$

2. $$0.0004352617 = 4.35 \times 10^{-4}$$

3. $$31,906,321,003 = 3.19\times 10^9$$

4. $$0.000000789654 = 7.90 \times 10^{-7}$$

5. $$301,888 =3.02 \times 10^5$$

6. $$0.010345 = 1.03\times 10^{-2}$$

7. $$978,909,456,001,745 = 9.79 \times 10^{14}$$

8. $$0.0050216 = 5.02\times 10^{-3}$$

9. $$8,927 = 8.93\times 10^3$$

10. $$0.435269 = 4.35\times 10^{-1}$$

Standard Form & Calculators

Although some calculators can present results in standard form, many use a slightly different notation to the one we have see here.

Calculators replace the number $$10$$ by the letter $$E$$

Examples

Here are some examples of what a calculator can display and the number, in standard form, they are referring to.

Example 1

With your calculator, try calculating: $987,654,321 \times 123,456,789$ You're likely to find: $1.21933\text{E}17$ This means: $1.21933\times 10^{17}$

Notice that we had to convert the number in decimal form, via:

Menu > Number > Convert to Decimal
to see it written in standard form $$1.21933\text{E}17$$

Example 2

With your calculator, try calculating: $5 \div 1234567890$ You're likely to see a result looking like: $4.05\text{E}-9$ This means: $4.05\times 10^{-9}$

Notice that we had to convert the number in decimal form, via:

Menu > Number > Convert to Decimal
to see it written in standard form $$4.05\text{E}-9$$

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