Activity 1.1.3 — Scientific & Engineering Notation¶

Learning Objectives¶

By the end of this lesson, students will be able to:

Convert numbers to and from scientific notation
Explain why engineering notation aligns with SI prefixes
Identify and use SI prefixes from pico to tera
Convert values between different SI prefixes
Apply notation skills to electronic component values

Vocabulary¶

Vocabulary (click to expand)

Term	Definition
Scientific notation	A number expressed as a x 10^n where 1 <= a < 10
Engineering notation	Scientific notation where the exponent is a multiple of 3
SI prefix	A prefix (kilo, mega, milli, etc.) that multiplies a base unit
Exponent	The power of 10 in scientific notation (n in a x 10^n)
Mantissa	The coefficient (a) in scientific notation

Part 1: Scientific Notation¶

Why Use Scientific Notation?¶

Digital electronics often involves very large or very small numbers:

A resistor might be 4,700,000,000 ohms
A capacitor might be 0.000000000001 farads
Current might be 0.000001 amps

Writing all these zeros is tedious and error-prone. Scientific notation makes these values easier to work with.

The Format¶

Scientific Notation:    a x 10^n

Where:
    a = mantissa (coefficient), where 1 <= a < 10
    n = exponent (integer, positive or negative)
    10 = base

Rules for Scientific Notation¶

The coefficient must be between 1 and 10 (not including 10)
The exponent tells you how many places to move the decimal point
Positive exponent = move decimal right (number gets larger)
Negative exponent = move decimal left (number gets smaller)

Examples¶

Standard Form	Scientific Notation	Explanation
3,400	3.4 x 10^3	Decimal moved 3 places left
0.0056	5.6 x 10^-3	Decimal moved 3 places right
150,000,000	1.5 x 10^8	Decimal moved 8 places left
0.00000042	4.2 x 10^-7	Decimal moved 7 places right

Converting to Scientific Notation¶

Step 1: Move the decimal point so there is only one digit to the left Step 2: Count how many places you moved it Step 3: If you moved right, exponent is negative; if you moved left, exponent is positive

Example: Convert 4,700,000 to scientific notation

Step 1: 4.7 (only one digit to left of decimal)
Step 2: Moved 6 places left (4,700,000 -> 4.7)
Step 3: Positive exponent

Result: 4.7 x 10^6

Part 2: Engineering Notation¶

The Problem with Scientific Notation¶

In engineering, we prefer exponents that are multiples of 3. This aligns with SI prefixes:

Multiple	Prefix	Symbol
10^3	kilo	k
10^6	mega	M
10^-3	milli	m
10^-6	micro	u

Engineering Notation Format¶

Engineering Notation:    a x 10^n

Where n is ALWAYS a multiple of 3

Comparing Notations¶

Standard	Scientific	Engineering
0.0047	4.7 x 10^-3	4.7 x 10^-3
4700	4.7 x 10^3	4.7 x 10^3
4700000	4.7 x 10^6	4.7 x 10^6

In engineering notation, you can also write these with SI prefixes: - 4700 ohms = 4.7 k ohms (or simply 4.7k) - 0.0047 farads = 4.7 mF (milli-farads)

Part 3: SI Prefixes Table¶

This table contains the most common prefixes used in electronics:

Prefix	Symbol	Factor	Decimal	Engineering Form
tera	T	10^12	1,000,000,000,000	1T
giga	G	10^9	1,000,000,000	1G
mega	M	10^6	1,000,000	1M
kilo	k	10^3	1,000	1k
(base unit)	-	10^0	1	1
milli	m	10^-3	0.001	1m
micro	u	10^-6	0.000001	1u
nano	n	10^-9	0.000000001	1n
pico	p	10^-12	0.000000000001	1p

Key insight: SI prefixes make numbers easier to read. Instead of 0.000001, we write 1u. Instead of 4,700,000, we write 4.7M.

Memory Aid for Large Prefixes¶

Think "Tiny Girls Microwave Pasta" for negative exponents going smaller: - Tera (10^12), Giga (10^9), Mega (10^6), Kilo (10^3)

For small prefixes, think "Most Monkeys Use Nuts Poorly" for increasing: - Milli (10^-3), Micro (10^-6), Nano (10^-9), Pico (10^-12)

Part 4: Converting Between SI Prefixes¶

Method 1: Move the Decimal¶

To convert between prefixes, move the decimal point in the opposite direction of the factor change:

Example 1: Convert 4.7k to plain ohms

4.7k = 4.7 x 10^3 = 4700 ohms
Moving from k to base: multiply by 10^3

Example 2: Convert 0.047A to mA

0.047A = ? mA
Moving from A (base) to milli: multiply by 10^3
0.047 x 10^3 = 47 mA

Method 2: Factor Analysis¶

Write the conversion as a ratio:

Example: Convert 2.2k ohms to ohms

2.2k ohms x (1000 ohms / 1k ohm) = 2,200 ohms

Example: Convert 4700 ohms to k ohms

4700 ohms x (1k ohm / 1000 ohms) = 4.7k ohms

Common Conversions in Digital Electronics¶

From	To	Multiply by
k ohms	ohms	1,000
M ohms	k ohms	1,000
ohms	k ohms	0.001
mA	uA	1,000
A	mA	1,000
uF	nF	1,000
nF	pF	1,000

Part 5: Worked Examples¶

Example 1: Converting Component Values¶

Problem: A resistor has the marking "4.7k". What is this value in ohms?

Solution:

4.7k = 4.7 kilo-ohms
     = 4.7 x 10^3 ohms
     = 4,700 ohms

Example 2: Scientific Notation to Engineering¶

Problem: Convert 8.2 x 10^6 hertz to engineering notation.

Solution:

8.2 x 10^6 Hz = 8.2 MHz
                = 8.2 megahertz

Example 3: Combining Operations¶

Problem: A capacitor is marked 0.047uF. What is this in pF?

Solution:

0.047uF = 0.047 x 10^-6 F
        = 4.7 x 10^-8 F
        = 4.7 x 10^4 x 10^-12 F
        = 4.7 x 10^4 pF
        = 47,000 pF

Example 4: Voltage Division¶

Problem: You have a 5V supply and need to express it in mV and kV.

Solution:

5V = 5 x 10^3 mV = 5,000 mV
5V = 5 x 10^-3 kV = 0.005 kV

Part 6: Practice Problems¶

Practice Problem 1 — Basic Conversion¶

Convert 2,700,000 ohms to engineering notation (k ohms and M ohms).

Show Solution

2,700,000 ohms = 2.7 x 10^6 ohms = 2.7M ohms
               = 2,700 x 10^3 ohms = 2,700k ohms

Practice Problem 2 — Scientific to Engineering¶

Express 3.3 x 10^-6 farads using the appropriate SI prefix.

Show Solution

3.3 x 10^-6 F = 3.3 uF (microfarads)

Note: 10^-6 is the micro (u) prefix

Practice Problem 3 — Back to Standard Form¶

What is 560nS in seconds (standard form)?

Show Solution

560nS = 560 x 10^-9 S
      = 5.60 x 10^2 x 10^-9 S
      = 5.60 x 10^-7 S
      = 0.00000056 seconds

Practice Problem 4 — Chain Conversion¶

Convert 0.15mA to uA and to A.

Show Solution

0.15mA to uA:
0.15mA = 0.15 x 10^-3 A
       = 150 x 10^-6 A
       = 150 uA

0.15mA to A:
0.15mA = 0.15 x 10^-3 A
       = 0.00015 A

Practice Problem 5 — Real World Component¶

A frequency is listed as 4.7 MHz. What is this in Hz and kHz?

Show Solution

4.7 MHz to Hz:
4.7 MHz = 4.7 x 10^6 Hz = 4,700,000 Hz

4.7 MHz to kHz:
4.7 MHz = 4.7 x 10^3 kHz = 4,700 kHz

Summary¶

Scientific notation: a x 10^n where 1 <= a < 10
Engineering notation: Exponent is a multiple of 3, aligns with SI prefixes
SI prefixes: T, G, M, k (large); m, u, n, p (small)
Conversion: Move decimal; direction depends on prefix magnitude
Memory aid: "Tiny Girls Microwave Pasta" for large; "Most Monkeys Use Nuts Poorly" for small

Key Reminders¶

SI prefixes make large and small numbers manageable
Engineering notation always uses exponents that are multiples of 3
When converting, move the decimal point in the direction that matches the factor change
Common in electronics: k ohms, M ohms, uF, nF, pF, MHz, GHz
Always include the unit when writing values (4.7k, not just 4.7)

Custom activity — adapted from PLTW Digital Electronics