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Scientific Notation Mathematics IMA
and Rounding Intro. to Fin. Maths I
To make life easier, we often rewrite large numbers in the form below:
30 , 000 = 3 × 10 , 000 = 3 × 104 235 , 600 = 2. 356 × 100 , 000 = 2. 356 × 105
where the decimal point (assumed to be at the end of these numbers) is “moved” to sit just after the first digit.
The same can be done for very small numbers such as 0.0018, if we observe the patterns in the following table of powers of ten:
10 = 101 (Anything to the power 1 is itself) 1 = 100 (Anything to the power 0 is 1) 1 10 ←^0.^1 =^10
− 1 1 100 ←^0.^01 =^10
− 2 1 1000 ←^0.^001 =^10
− 3 .. .
Each time we move down a line, the decimal point moves one step to the left and the power of ten reduces by one. Hence, negative powers of 10 have a meaning as decimal parts of numbers. For example:
= 1. 8 ×
= 1. 8 × 10 −^3.
This alternative form is called scientific notation and is based on the fact that moving the decimal point to the left or right in a number is equivalent to multiplying the number by a power of 10 (negative powers amount to dividing).
Any number can be represented using scientific notation (although we don’t usually write numbers between 0 and 10 as 0 × 100 , 1. 25 × 100 , etc.); in particular, this is the type of notation used by calculators for numbers with more digits in them than can be displayed.
To change a number from ordinary decimal notation to scientific notation, shift the decimal point so that it lies on the immediate right of the first non-zero digit of the number and multiply it by the appropriate power of 10.
Here are some examples:
(a) 69 , 000 , 000 = 6. 9 × 107 (b) 0 .0000000837 = 8. 37 × 10 −^8
(c)
= 0.25 = 2. 5 × 10 −^1 (d)
= 0.002¯7 = 2.¯ 7 × 10 −^3
(e)
= 133.¯3 = 1.¯ 3 × 102 (f) 3. 1762 × 10 −^2 = 3. 1762 ×
Exercise
(1) Convert to Scientific notation or vice versa:
(a) 3 , 800 , 000 = (b) = 2. 9 × 104
(c) 0. 000019 = (d) = 1. 57 × 10 −^4
(e)
= (f)
Rounding Numbers and Significant Figures
In mathematics, the exact solution to a calculation may involve many (if not infinitely many) decimal places. For example, if we wanted to make a rectangular frame where one side was 1 metre in length and the other had to be two thirds of this, the exact length required would be 0. 6666... metres. For practical purposes we would probably only need to measure “to the nearest millimetre” (0.667 metres).
The number 0.667 is said to have three significant figures and we have rounded off at the third decimal place. Significant figures are all digits present in a number with the exception of ‘place marking’ or leading zeros in purely decimal numbers. Here are some other examples:
Number Significant Digits
- 023 5
- 00019 2
- 00019 6
- 000 4
The leading zeros in the second example vanish if you write 0.00019 as 1. 9 × 10 −^4. Notice that ‘trailing zeros’ are important as these indicate the level of accuracy of the measurement as much as any other digits.
Answers to Exercises
- (1) (a) 3. 8 × 106 (b) 29, 000 (c) 1. 9 × 10 −^5 (d) 0.
- (e) 0.875 = 8. 75 × 10 −^1 (f) 2. 3 ×
- (2) (a) 3. 280 × 10 −^3 , 4 (b) 7. 9900106 × 106 , 8 (c) 1. 8004 × 101 , 5 (d) 9. 851 × 10 −^7 ,
- (e) 8. 1418414900 × 108 ,
- (3) (a) 3. 3 × 10 −^3 (b) 8. 0 × 106 (c) 1. 8 × 101 (d) 9. 9 × 10 −
