Know and apply Standard form, significant figures, rounding and decimal place value [L5]
Scientific notation or standard form as it is sometimes called, is a useful and shorthand method for writing very large or very small numbers. As the name suggests, scientific notation is used in the area of science but may also be found in mathematics, engineering and medicine. In fact just about anywhere where large or minute quantities need to be counted you will find the short hand use of scientific notation in use. Most of the examples used here come from astronomy and chemistry because space is a huge place and the distance between atoms is incredibly small. You might find the table below useful to understand the size of some measurements. Examples of large numbers might be the number of atoms in the human body or the mass of the earth in kg. A small number might be the mass of an atom. You could also revisit the 'Scale of the Universe'. The link from Google Classroom is still there.
Converting from normal notation to scientific notation
Let's have a look at some special numbers in physics and chemistry.
The first is the speed of light
300,000,000 m/s
The second number is called Avogadro's number. It's pretty big!
602,200,000,000,000,000,000,000 atoms in 12 grams of Carbon.
Commas have been used to make counting the decimal places easier. In scientific notation the number is expressed as the first digit or non zero digits if there is more than one digit at the start, multiplied by a power of 10 that indicates the number of decimal places to the end of that number, shown as 'x10 to the power of a number', where the number is the number of decimal places. Unfortunately website structure will not allow the scientific notation to be shown as powers. So check out the video or image file below.
The first is the speed of light
300,000,000 m/s
The second number is called Avogadro's number. It's pretty big!
602,200,000,000,000,000,000,000 atoms in 12 grams of Carbon.
Commas have been used to make counting the decimal places easier. In scientific notation the number is expressed as the first digit or non zero digits if there is more than one digit at the start, multiplied by a power of 10 that indicates the number of decimal places to the end of that number, shown as 'x10 to the power of a number', where the number is the number of decimal places. Unfortunately website structure will not allow the scientific notation to be shown as powers. So check out the video or image file below.
Scientific Notation
Activity 5. Write these numbers in scientific notation.
1) 470
2) 5000
3) 60
4) 3600
5) 7000000000
6) 15
7) 6.8
8) 890000
9) 365
10) 620000
Activity 5. Write these numbers in scientific notation.
1) 470
2) 5000
3) 60
4) 3600
5) 7000000000
6) 15
7) 6.8
8) 890000
9) 365
10) 620000
Convert the measurements in the following exercise from scientific notation to normal or ordinary numbers by writing out the digits first and then moving the decimal point, adding zeros specified by the power expressed.
Converting very small values
Scientific Notation
Activity 6
Exercises: Write these numbers in scientific notation using negative powers of 10
1) 0.4
2) 0.0023
3) 0.045
4) 0.9
5) 0.83
6) 0.006
7) 0.00000002
8) 0.0312
9) 0.204
10) 0.00081
Activity 6
Exercises: Write these numbers in scientific notation using negative powers of 10
1) 0.4
2) 0.0023
3) 0.045
4) 0.9
5) 0.83
6) 0.006
7) 0.00000002
8) 0.0312
9) 0.204
10) 0.00081
Scientific Notation
Activity 7
Convert these numbers from Scientific to ordinary numbers
Activity 7
Convert these numbers from Scientific to ordinary numbers
Find Powers [L5]
powers of negative numbers [l6]
Negative numbers can also be written in power form. The rules of 'negative x negative = positive' and 'negative x positive = negative' apply. Let's have a look at an example.
Extend Powers to include Square Roots (L5)
What number when multiplied by itself gives 25?
i.e. 25 = ? x ? The answer is 5 because 5 x 5 = 25
What number when multiplied by itself gives 9?
i.e. 9 = ? x ? The answer is 3 because 3 x 3 = 9
We say the square root of 25 is 5 and the square root of 9 is 3. This can be expressed mathematically as:
√25 = 5 or √9 = 3
(At this point it is worth noting that -5 x -5 also gives 25
And -3 x -3 also gives 9) so technically √25 = ±5 and
√9 = ±3, but don't worry too much about that just yet.
i.e. 25 = ? x ? The answer is 5 because 5 x 5 = 25
What number when multiplied by itself gives 9?
i.e. 9 = ? x ? The answer is 3 because 3 x 3 = 9
We say the square root of 25 is 5 and the square root of 9 is 3. This can be expressed mathematically as:
√25 = 5 or √9 = 3
(At this point it is worth noting that -5 x -5 also gives 25
And -3 x -3 also gives 9) so technically √25 = ±5 and
√9 = ±3, but don't worry too much about that just yet.
Extra: cube roots.
What 'same' number when multiplied by itself 3 times gives 8?
i.e. 8 = ? x ? x ? The answer is 2 because 2 x 2 x 2 = 8
What number when multiplied by itself 3 times gives 27?
i.e. 27 = ? x ? x ? The answer is 3 because 3 x 3 x 3 = 27
We say the cube root of 8 is 2 and the cube root of 27 is 3. This can be expressed mathematically as:
3√8 = 2 or 3√27 = 3 (technical glitch - the 3 at the front of the root symbol should be a little higher!)
Evaluate these cube roots
1) ³√27
2) ³√125
3) ³√8
4) ³√64
5)³√ 1000
i.e. 8 = ? x ? x ? The answer is 2 because 2 x 2 x 2 = 8
What number when multiplied by itself 3 times gives 27?
i.e. 27 = ? x ? x ? The answer is 3 because 3 x 3 x 3 = 27
We say the cube root of 8 is 2 and the cube root of 27 is 3. This can be expressed mathematically as:
3√8 = 2 or 3√27 = 3 (technical glitch - the 3 at the front of the root symbol should be a little higher!)
Evaluate these cube roots
1) ³√27
2) ³√125
3) ³√8
4) ³√64
5)³√ 1000
Using the calculator to solve powers and roots
The calculator can be used to carry out a variety of power and root functions (and other things). Get to know your calculator well. Although the number assessment is calculator free you can still use it to check your answers during the practises as you go.
negative indices [L6]
A power with a negative index is the same as the reciprocal of the same power with a positive index: