Standard Form

Powers: 822223 255552

Revision

On calculator: 8 3 223 yx

2 is multiplied by itself 3 times

Try these: 65 54 36 25 94 104

Answers: 7776 625 729 32 6561 10 000

3210123 10101010101010

Look at these sequences

1000

1

100

1

10

11101001000

3210123

10

1

10

1

10

110101010

Powers of 10

Powers of 10 are easy e.g.103 = 10 x 10 x 10 =1000

3 zeros

001.001.01.01101001000

Divide by 10

From this we see that 1100 1.010 1 01.010 2

Working with big numbers

Scientists often have to work with big numbers.

The distance the earth travels in one orbit is 558 000 000 miles.

We can express this number in a more concise form as follows.

Example:

558 000 000 = 5.58 x 100 000 000 = 5.58 x 108

is said to be in Standard Form

or Scientific Notation

5.58 x 108

558 000 000 is said to be in Normal Form

Look at this number which is in Standard Form

6.3 x 105

exactly one digit before the point

the power is a positive or negative whole number

6.3 x 105 = 6.3 x 10 x 10 x 10 x 10 x 10

= 630 000

Standard form Ordinary Numberwatch this

6.3 63.

1

630.

2

6300.

3

63000.

4

630000.

5

x 105

5 jumps

Example: Write 2.45 x 104 as an ordinary number.

Method: • Write the question

• List the digits without the point

• 4 jumps from position of “old” point

• Write the answer

2 4 500

00

Add zeros as required

1234Now try these:

Write as ordinary numbers:

1.47 x 102 9.08 x 106 1.3 x 100

4 x 103 4.88 x 101

147 9080000 1.3 4000 48.8

Standard formOrdinary Number watch this

= 6.3 x 105= 6.3 x 100 000630 000

6.363.630.

2

6300.

3

63000.

4

630000.

1 5

x 105

5 jumps

00 000 00000 0 0 0 0

Therefore

630 000 = 6.3 x 105

Example: Write 2706 in standard form.

Method: • Write the number

• Insert “new” decimal point

• Count jumps to position of “old” decimal point2 7 0 6 123

Now try these:

Write in standard form:

34560 1023.6 12.8 4.6 230000

3.456 x 104 1.0236 x 103 1.28 x 101 4.6 x 100 2.3 x 105

= 2 7 0 6 x 10 3

Positive power since

large number

Make the number “look” like standard form

Put in a decimal point to make the number “look” like a number between 1 and 10.

Numbers less than 1

Consider the number 2.03 x 10-3

This number is in standard form but is different from previous examples.

exactly one digit before the point

the power is a positive or negative whole number

Notice the power is negative

2.03 x 10-3 = 2.03 x 0.001 = 0.00203

Look again at powers

of 10

Standard form Ordinary Numberwatch this

2.03 x 10-3.203 .0203 .00203

x 10-3

3 jumps

1 2 3

= 0.00203

Example: Write 1.07 x 10-4 as an ordinary number.

Method: • Write the question

• List the digits without the point

• 4 jumps from position of “old” point

• Write the answer

1 0 700

00

Add zeros as required

1234

Now try these:Write as ordinary numbers:

1.47 x 10-2 9.08 x 10-5 1.3 x 10-1 4 x 10-3 4.081 x 10-4

0.0147 0.0000908 0.13 0.004 0.0004081

000

That means the answer starts with 0.-----

Note: When the power is negative the ordinary number is always less than 1.

Standard formOrdinary Number watch this

= 6.31 x 10-3= 6.31 x 0.0010.00631

6.31

2

0.631

3

0.06310.00631

1

x 10-3

3 jumps

0000 0 0

Therefore

630 000 = 6.31 x 10-3

Example: Write 0.0027 in standard form.

Method: • Write the number

• Insert “new” decimal point

• Count jumps to position of “old” decimal point

= 2 7

123

Now try these:

Write in standard form:

0.3456 0.00102 0.0128 0.000046 0.0000000023

3.456 x 10-1 1.02 x 10-3 1.28 x 10-2 4.6 x 10-5 2.3 x 10-9

Put in a decimal point to make the number “look” like a number between 1 and 10.

0 0 0 2 7

x 10

Negative power since tiny number -

3

Make the number “look” like standard form

Standard Form on the Calculator

We use EXP button as shown on the calculator to enter numbers already in standard form.

Examples:

2.6 x 108 2 6 EXP 8 =

260 000 000check this is

correct

3.5 x 1012

3.5 12

3 5 EXP 1 2 =

Too large to convert

Calculator’s way of writing 3.5 x 1012 – does not mean 3.5 to the power 12!

Calculations Giving your Answer in Standard Form

Example 1: (2.6 x 108) x (4.2 x 106) calculator display

1.092 15= 1.092 x 1015

Example 2: calculator display

5.4 10= 5.4 x 1010

(2.484 x 107) (4.6 x 10-

4)

Note:

• Only enter EXP before power, never x 10 EXP

• For negative powers use the ( - ) key not the key, we are not subtracting.

• Remember to change number on display to ---- x 10--

See calculators

again

Click button to end presentation

( - )

END