Introduction

Engineering notation is a quick way of converting a quantity as represented on a calculator to a prefixed notation. Engineering notation is similar to Scientific Notation, except the power of 10 is always a multiple of 3. The multiple of 3 is handy, since it is a direct mapping between powers of 10 and the unit prefix.

It is also possible to do any one of these problems by hand, by direct inspection of the number.


Scientific Notation and Engineering Notation Form

The forms of Scientific Notation and Engineering Notation are given below.

  • Scientific Notation: \( (\pm) m \times 10^{n} \), where \( \pm\) represents the sign of the number (whether it is positive or negative), \( m \) is the mantissa or significand, and \( n \) is the exponent. The significand \( m \) is any value between 1 and just less than 10, and \( n \) is always an integer.

  • Engineering Notation: \( (\pm) p \times 10^{3q} \), where \( \pm\) represents the sign of the number (whether it is positive or negative), \( p \) is the mantissa or significand, and \( q \) is the exponent. The significand \( p \) is usually any value between 1 and just less than 1000, and \( q \) is always an integer.


Converting from Scientific Notation to Engineering Notation

First, check the exponent. If the exponent is already a multiple of 3, then no action is necessary.

  • 4.99 × 10-9 is the same in both Scientific Notation and Engineering Notation.
  • 671 is the same in both Scientific Notation and Engineering Notation.
  • 6.2 × 106 is the same in both Scientific Notation and Engineering Notation.

If the exponent is not a multiple of 3, then it needs to be converted to the next multiple of 3 closer to negative infinity.

The significand needs to be multiplied so that the overall value is kept the same.

The table below shows these conversions, and the unit prefix.

Rather then memorising this table, it is recommended that you memorise the patterns.

Scientific
Notation
Exponent
 Engineering
Notation
Exponent
 Engineering
Notation
Multiplier
 Unit Prefix
14 12 100 T
13 12 10 T
12 12 1 T
11 9 100 G
10 9 10 G
9 9 1 G
8 6 100 M
7 6 10 M
6 6 1 M
5 3 100 k
4 3 10 k
3 3 1 k
2 0 100 -
1 0 10 -
0 0 1 -
-1 -3 100 m
-2 -3 10 m
-3 -3 1 m
-4 -6 100 µ
-5 -6 10 µ
-6 -6 1 µ
-7 -9 100 n
-8 -9 10 n
-9 -9 1 n
-10 -12 100 p
-11 -12 10 p
-12 -12 1 p


The example in the sheet worksheet is 2.2 × 105 V.

Looking at the table above, for a Scientific Notation exponent of 5, the following is found:

  • The Engineering Notation exponent is 3.
  • The unit prefix is k.
  • The Engineering Notation multiplier is 100.

So, 2.2 × 105 V in engineering notation is (2.2 × 100) × 103 V = 220 × 103 V, or (2.2 × 100) kV = 220 kV.


Converting from Engineering Notation to Scientific Notation

First, check the exponent. If the exponent is already a multiple of 3, and the significand is between 1 and 10, then no action is necessary.

  • 4.99 × 10-9 is the same in both Engineering Notation and Scientific Notation.
  • 6.2 × 106 is the same in both Engineering Notation and Scientific Notation.

If the exponent is not a multiple of 3, then it needs to be converted to the next multiple of 3 closer to negative infinity.

The significand needs to be multiplied so that the overall value is kept the same.

The table below shows these conversions, and the unit prefix.

Rather then memorising this table, it is recommended that you memorise the patterns.

Engineering
Notation
Exponent
 Unit Prefix
 Engineering
Notation
Range
 Scientific
Notation
Multiplier
 Scientific
Notation
Exponent
12 T 100-999.9… 0.01 14
12 T 10-99.9… 0.1 13
12 T 1-9.9… 1 12
9 G 100-999.9… 0.01 11
9 G 10-99.9… 0.1 10
9 G 1-9.9… 1 9
6 M 100-999.9… 0.01 8
6 M 10-99.9… 0.1 7
6 M 1-9.9… 1 6
3 k 100-999.9… 0.01 5
3 k 10-99.9… 0.1 4
3 k 1-9.9… 1 3
2 - 100-999.9… 0.01 0
1 - 10-99.9… 0.1 0
0 - 1-9.9… 1 0
-3 m 100-999.9… 0.01 -1
-3 m 10-99.9… 0.1 -2
-3 m 1-9.99… 1 -3
-6 µ 100-999.9… 0.01 -4
-6 µ 10-99.9… 0.1 -5
-6 µ 1-9.9… 1 -6
-9 n 100-999.9… 0.01 -7
-9 n 10-99.9… 0.1 -8
-9 n 1-9.99… 1 -9
-12 p 100-999.9… 0.01 -10
-12 p 10-99.9… 0.1 -11
-12 p 1-9.9… 1 -12


The example in the sheet worksheet is 220 × 103 V.

Looking at the table above, for an Engineering Notation exponent of 3, the following is found:

  • The Engineering Notation exponent is 3.
  • The unit prefix is k.
  • The Engineering Notation Range multiplier is 100-999.9….
  • The Scientific Notation Multiplier is 0.01.
  • The Scientific Notation Exponent is 5.

So, 220 × 103 V in Scientific Notation is (220 × 0.01) × 105 V = 2.2 × 105 V.


Calculator - Using ENG Function

The worksheet in the book is oriented towards a Casio FX-82MS. Your calculator may have a different method of working with Engineering Notation.

It is also possible to convert to different forms of notation by direct inspection of the output, by counting digits.

You will need to work out how your calculator works with Scientific Notation, and Engineering Notation.


Calculator - Using Logarithms

Another calculator-based method is to use base-10 logarithms. The base-10 logarithm function calculates the value \( x \) for a value \( y \) such that \( 10^{x} = y \). In this case, \( x = \log y \), where \( log \) represents the base-10 logarithm function.

The logarithm enables you to calculate the exponent for scientific notation.

To do this, you take the logarithm of the number (\( \log x \)), and round the number towards negative infinity, also known as the floor function. Rounding towards negative infinity means that a positive number like 5.5 gets rounded down to 5. Negative numbers like -5.5 get rounded to -6, since -6 is closer to negative infinity then -5.5.

With that in mind, the floor function of the logarithm of the number gives the exponent for scientific notation.

The mantissa or significand (the digits in the scientific notation) are the same as the digits in the original number, except the decimal point is moved to just after the first non-zero digit.

Example 1: 0.00356

  • The exponent: \( \log 0.00356 = -2.44\ldots \) i.e. the exponent is -3.
  • The significand is 3.56.

With that known, 0.00356 = 3.56 × 10-3 in scientific notation.

Example 2: 35600000

  • The exponent: \( \log 35600000 = 7.55\ldots \) i.e. the exponent is 7.
  • The significand is 3.56.

With that known, 35600000 = 3.56 × 107 in scientific notation.


Scientific Notation By Direct Inspection

Scientific notation can also be determined by direct inspection.

The mantissa or significand (the digits in the scientific notation) digits are the same as the digits in the original number, except the decimal point is moved to just after the first non-zero digit.

The exponent is determined by digit counting.

If a number is more than 1, and has \( n + 1 \) digits before the decimal point, then the Scientific Notation exponent is \( n \).

If a number is less than 1, and has \( n - 1 \) leading zeros after the decimal point, then the Scientific Notation exponent is \( -n \).

Examples:

  • 314159.265: There are six digits before the decimal point, so the Scientific Notation exponent is 5. The mantissa is 3.14159265, so the Scientific Notation is 3.14159265 × 105.

  • 0.0000000002718: There are nine leading zeros after the decimal point, so the Scientific Notation exponent is -10. The mantissa is 2.718, so the Scientific Notation is 2.718 × 10-10.


Engineering Notation By Direct Inspection

Engineering notation can also be determined by direct inspection.

If a number is more than 1, group the digits in threes starting left from the decimal point, until there are no digits left. While the number left of the decimal point is more than or equal to 1000, move the decimal point 3 places left, and add 3 to the exponent. The significand is the left-most group of digits, with the decimal point on the right. The rest of the digits to the right are appended on with any other decimal points removed.

If a number is less than 1, group the digits in threes starting right from the decimal point, until there is at least one non-zero digit in the group. While the number right of the decimal point is less than 1, move the decimal point 3 places right, and subtract 3 from the exponent. The significand is the first group of digits, with the decimal point on the right. The rest of the digits to the right are appended on with any other decimal points removed.

Examples:

  • 314159.265: The grouping is 314 159.265. The decimal point is moved left once, to produce 314.159265. The decimal point is moved left once, so the Engineering Notation exponent is 3. The Engineering Notation version of 314159.265 is therefore 314.159265 × 103.

  • 0.0000000002718: The grouping is 0.000 000 000 271 8. The decimal point is moved right four times, to produce 271.8. The decimal point is moved right four times so the Engineering Notation exponent is -12. The Engineering Notation version of 0.0000000002718 is therefore 271.8 × 10-12.

Assignment Answers and Commentary

The answers here are presented in bold.

Some notes:

  1. Ordinary Number and Appropriate form are subjective. For example capacitance (farad, F) is more often written in pF, or µF, even when other prefixes could be used. For example, 4700 µF is used more often than 4.7 mF, and capacitances between 1 nF and 1 µF are often given in pF or µF e.g. 0.047 µF instead of 47 nF, or 1200 pF (or 0.0012 µF) instead of 1.2 nF.

  2. Normally, the exponent in Engineering Notation is never greater than the exponent in Scientific Notation.

  3. The mantissa (or significand) of a Scientific Notation or the significand of an Engineering Notation number is normally at least 1.


Ordinary Number
 Scientific Notation
 Engineering Notation
 Appropriate Form
 Comment
220000 V
 2.2 × 105
 2.2 × 103
 220 kV
0.0003 Ω
 3 × 10-4 Ω
 300 × 10-6 Ω
 300 µΩ
 This may be analysed as 0.0003 Ω → 0.3 mΩ → 300 µΩ. The 'µ' instantly implies an Engineering Notation exponent of -6.

The Scientific Notation is obtained by multiplying the 10-6 by 100 and dividing the significand by 100 so the significand is between 1 and 10.
0.000567 5.67 × 10-4
 567 × 10-6
 567 µ The '-4' on the scientific notation instantly implies three zeros after the decimal point on the Ordinary Number.

The Engineering Notation can obtained by going 5.67 × 10-4 → 56.7 × 10-5 → 567 × 10 -6.
17000 A
 1.7 × 104 A
 17 × 103 A
 17 kA
 The 'k' prefix instantly implies that the Engineering Notation multiplier is 3.

The Scientific Notation is obtained by multiplying the 103 by 10 and dividing the significand by 10 so the significand is between 1 and 10.
65000000000 W
 6.5 × 1010 W
 		65 × 109 W
 65 GW
 The Ordinary Number has 11 digits before the (implicit) decimal point, which implies that the unit prefix must be 'G', since G means 1 billion, which suits numbers with between 9 and 11 digits before the (implicit) decimal point. The 'G' prefix gives us '9' for the Engineering units exponent.

The Scientific Notation is obtained by division-multiplication until the significand is between 1 and 10 i.e.  65 × 109 → 6.5 × 1010.
0.00000022 2.2 × 10-7
 220 × 10-9 220 n The scientific notation exponent -7 implies six leading zeros after the decimal point.

The Engineering Notation and "Approriate Form" can be found by noting that the nearest multiple of 3 less than or equal to -7 is -9. Therefore, the Engineering Notation exponent is -9, and the prefix is 'n' (nano). Since the exponent 10-9 is 100 times less than 10-7, the significand must be 100 times more, to keep the value the same.
33000000 V
 3.3 × 107 V 33 × 106 V 33 MV
 There are 8 digits in this number before the (implicit) decimal point. This implies that the prefix should be M (mega), and the Engineering Notation exponent is 6.

The significand on the Engineering Notation form is 33. To get this into Scientific Notation, the significand must be between 1 and 10. Dividing 33 by 10, and multiplying the exponent by 10 will do the trick.
0.000092 A
 9.2 × 10-5 A 92 × 10-6 A 92 µA
 The 'µ' prefix instantly implies that the Engineering Notation multiplier is -6.

The Scientific Notation is obtained by multiplying the 10-6 by 10 and dividing the significand by 10 so the significand is between 1 and 10, so that the value is preserved.

The Scientific Notation exponent -5 implies four leading zeros after the decimal point.
170000 1.7 × 105
 170 × 103 170 k The Scientific Notation exponent 5 implies that the significand digits are to be zero padded until the Ordinary Number notation has six digits before the (implicit) decimal point. In this case, four zeros are required since '17' is two digits.

The Engineering Notation and "Appropriate Form" can be found by noting that the nearest multiple of 3 less than or equal to 5 is 3. Therefore, the Engineering Notation exponent is 3, and the prefix is 'k' (kilo). Since the exponent 103 is 100 times less than 105, the significand must be 100 times more, to keep the value the same.
0.000056 F
 5.6 × 10-5 F 56 × 10-6 F 56 µF The four leading zeros after the decimal point in the Ordinary Number instantly implies a scientific notation exponent of -5.

The Engineering Notation and "Appropriate Form" can be found by noting that the nearest multiple of 3 less than or equal to -5 is -6. Therefore, the Engineering Notation exponent is -6, and the prefix is 'µ' (micro). Since the exponent 10-6 is 10 times less than 10-6, the significand must be 10 times more, to keep the value the same.
0.000000002 2 × 10-9 2 × 10-9
 2 n
 In this case, the significand on the Engineering Notation is already between 1 and 10. Therefore, the Scientific Notation is the same as the Engineering Notation.

The Engineering Notation exponent -9 instantly implies a prefix of 'n' (nano).

The Ordinary Number is obtained by noting that a Scientific Notation exponent of 9 implies eight leading zeros after the decimal point.
100000000000000 1 × 1014 Bytes 100 × 1012 Bytes
 100 TBytes
 The 'T' prefix instantly implies that the Engineering Notation multiplier is 12.

The Scientific Notation is obtained by multiplying the 1012 by 100 and dividing the significand by 100 so the significand is between 1 and 10, so that the value is preserved.

The Scientific Notation exponent 14 implies that the Ordinary Number has 15 digits before the (implicit) decimal point. Since '1' is a single digit, we need to add 14 trailing zeros to get the Ordinary Notation.
400 4 × 102 400 400 The Scientific Notation exponent 2 implies that the Ordinary Number has 3 digits before the (implicit) decimal point. Since '4' is a single digit, we need to add 2 trailing zeros to get the Ordinary Notation.

The Engineering Notation and "Appropriate Form" can be found by noting that the nearest multiple of 3 less than or equal to 2 is 0. Therefore, the Engineering Notation exponent is 0, and the prefix is not required.

The Engineering Notation could be written 476 × 100. However, 100 = 1, so it is redundant.In fact, since the Ordinary Number is between 1 and 1000, the Engineering Notation and  Appropriate Form don't need to be changed from the Ordinary Number form.
476 A
 4.76 × 102 A
 476 A
 476 A
 Since the Ordinary Number is between 1 and 1000, the Engineering Notation and Appropriate form don't need to be changed from the Ordinary Number form.

The Scientific Notation is obtained by noting that 476 is between 100 and 1000. Therefore, the exponent is 102. The significand is obtained by dividing 476 by 100. so that the value is preserved.
0.0000000009 C
 9 × 10-10 C
 900 × 10-12 C
 900 pC
 The nine leading zeros after the decimal point instantly implies a scientific notation exponent of -10.

The Engineering Notation and "Appropriate Form" can be found by noting that the nearest multiple of 3 less than or equal to -10 is -12. Therefore, the Engineering Notation exponent is -12, and the prefix is 'p' (pico). Since the exponent 10-12 is 100 times less than 10-10, the significand must be 100 times more, to keep the value the same.
5000000000000
 5 × 1012
 5 × 1012
 5 T
 In this case, the significand on the Engineering Notation is already between 1 and 10. Therefore, the Scientific Notation is the same as the Engineering Notation.

The Engineering Notation exponent 12 instantly implies a prefix of 'T' (tera).

The Scientific Notation exponent 12 implies that the Ordinary Number has 13 digits before the (implicit) decimal point. Since '5' is a single digit, we need to add 12 trailing zeros to get the Ordinary Notation.


آخر تعديل: الاثنين، 11 مايو 2020، 4:55 PM