B2B: Scientific Notation
Do you know the difference between Standard Notation and Scientific Notation?
When a number is written using only numbers.
Example
65 890
0.0000397
Scientists frequently deal with numbers that are both very small and very big. For example, the diameter of a living cell is microscopic, while distances in space are immense. Using Standard Notation would involve too many zeros and could lead to confusion or calculation error. Thus, representing these numbers as powers of 10 is often the best practice.
Scientific Notation takes the form:
M x 10n where 1 ≤ M < 10, and n = the number of decimal places to be moved
**The exponent value, n, is equal to the number of decimal points moves (or "jumps") because both represent a factor of ten**
Importantly,
- a positive n indicates that the standard form is a large number
- a negative n indicates that the standard form is a small number
1. The speed of light in a vacuum is 299 792 458 metres per second. When rounded, this is approximately: 3 x 108 m/s. (We move the decimal point so there is only one digit to its left, a total of 8 places). Press the numbers (from right to left) to see how the decimal moves.
= 3 x 108
Remember: a positive n indicates that the standard form is a large number
10 8
10 7
10 6
10 5
10 4
10 3
10 2
10 1
Non-decimal numbers
← Numbers move from the right to the left
2. Small lymphocytes (white blood cells) are approximately 0.000007 metres in diameter. This is approximately: 7 x 10-6 m (Here we move the decimal point to the right, 6 places). Press the numbers (from left to right) to see how the decimal moves.
= 7 x 10-6
Remember: a negative n indicates that the standard form is a small number
10 -1
10 -2
10 -3
10 -4
10 -5
10 -6
Decimal numbers
Numbers move from the left to the right →
Please look at the Scientific Number Line. Select and move the slider to see how the magnitude of a number changes as you move to the left and right. Make note of the corresponding Standard Notation values.
Two things can be activated as the slider moves:
- the corresponding number in Standard Notation can appear above (0.00001 to 100 000)
- a graphic, ie. a donut, will appear in relative number – from a speck of a piece, through to one whole donut at 1 x 100 (=1), and over to a massive pile of donuts at the far right of the number line.
- Try converting the following "exact" numbers between Scientific Notation (M x 10n ) and Standard Notation:
a) 93 476 g
x 10
You’re right!
Not quite. Let’s take a look at that again.
The correct answers are displayed.
For ‘M’ to be 1 or greater, and smaller than 10, the decimal moves 4 places to the left. The exponent is (+) 4 (positive), indicating that the number is large.
- Try converting the following "exact" numbers between Scientific Notation (M x 10n ) and Standard Notation:
b) 0.00391 s
x 10
You’re right!
Not quite. Let’s take a look at that again.
The correct answers are displayed.
The decimal moved 3 places to the right so the exponent is (-) 3 (negative). Whenever a number starts out less than 1, the exponent is always negative.
- Try converting the following "exact" numbers between Scientific Notation (M x 10n ) and Standard Notation:
c) 4 x 101
You’re right!
Not quite. Let’s take a look at that again.
The correct answers are displayed.
Remember 100 = 1
- Please put your understanding to practice by performing the following calculations, in scientific notation:
a) (1.5 x 103 ) x (3.5 x 105 )
x
You’re right!
Not quite. Let’s take a look at that again.
The correct answers are displayed.
The first parts are multiplied together and the exponents are added together.
- Please put your understanding to practice by performing the following calculations, in scientific notation:
b) (63.1 x 1012 ) ÷ (1.1 x 103 )
x
You’re right!
Not quite. Let’s take a look at that again.
The correct answers are displayed.
The first parts are divided and the exponents are subtracted. We rounded to 2 decimal places.
- Please put your understanding to practice by performing the following calculations, in scientific notation:
c) (4 x 105 ) ÷ (0.5 x 10-3 )
x
You’re right!
Not quite. Let’s take a look at that again.
The correct answers are displayed.
The first parts are divided and the exponents are subtracted, 5 – (-3) = 5 + 3 = 8.
Think back to the number line slider you saw earlier. It is shown here but has been modified to include unit prefixes. Both systems of naming are based on powers of ten.
