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Math help from the Learning Centre

This guide provides useful resources for a wide variety of math topics. It is targeted at students enrolled in a math course or any other Centennial course that requires math knowledge and skills.

Learning Outcomes

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

  1. Distinguish between scientific notation, engineering notation, and prefix notation.

  2. Convert numbers and measurements among the three systems.

  3. Recognize when each notation is most useful in scientific and engineering contexts.

  4. Reflect on common misconceptions and errors to strengthen productive habits of checking and noticing.

Introduction

Imagine you’re an engineer measuring a capacitor that is 0.000000047 farads. How should you report this?

  • In scientific notation: \(4.7\times10^{−8}\)

  • In engineering notation: \(47\times10^{−9}\)

  • In prefix notation: \(47nF\)

All three are mathematically equivalent, but each highlights different aspects. This lesson will help you understand why and how to use them.

Core Concepts with Variation

A. Scientific Notation

  • Any number written as \(a\times10^{n}\), where \(1\leq a \leq10\).

  • Example: \(3.2\times 10^{5}\).

  • Variation focus: Exponents can be any integer.

B. Engineering Notation

  • Similar, but exponents must be multiples of 3.

  • Example: \(320\times 10^{3}=3.20 \times 10^{5}\).

  • Variation focus: Notice how values are shifted to match powers of 10 divisible by 3.

C. Prefix Notation

  • Uses SI prefixes (k, M, µ, n) that align with multiples of 3 exponents. (Table of SI prefixes below)

  • Example: \(320\times 10^{3} V=320 kV\).

  • Variation focus: Emphasize correspondence: k → \(10^3\), M → \(10^6\), µ → \(10^P{−6}\).

Worked Examples

Example 1: \(0.00056m\)

  • Scientific: \(5.6 \times 10^{-4}\)m
  • Engineering: \(560 \times 10^{-6}\)m
  • Prefix: 560 \(\mu\)m.

Variation emphasis: See how the same quantity is represented differently.

Example 2: \(3.2 \times 10^{8}\)Hz

  • Scientific: \(3.2 \times 10^{8}\)Hz
  • Engineering: \(320 \times 10^6\)Hz
  • Prefix: 320 MHz

Variation emphasis: Exponent flexibility vs. engineering multiples of 3 vs. practical communication.

Productive Struggle Tasks

Task 1: Spot the Error

A student writes: 

\[2.5mA=2.5 \times 10^{-6}A\]

  • Is this correct? If not, correct it.

  • Try writing the value in all three notations.

Task 2: Conversion Challenge

Convert \(6.8 \times 10^{-9}\)s into:

  • Scientific notation

  • Engineering notation

  • Prefix notation

Task 3: Application Problem

The speed of light is \(3.0\times 10^8\) m/s.
How far does light travel in \(7.5\times 10^{−9}\)s?

  • Compute the distance.

  • Express the answer in scientific, engineering, and prefix notation.

Reflection Prompts

  • When is scientific notation more useful than prefix notation?

  • Why do engineers prefer engineering notation or prefixes in circuit design?

  • Which system do you find easiest to misapply? Why?

SI Prefixes

Name Symbol Multiplying Facotr
tera \(T\) \(10^{12}\)
giga \(G\) \(10^9\)
mega \(M\) \(10^6\)
kilo \(k\) \(10^3\)
hecto \(h\) \(10^2\)
deca \(da\) \(10^1\)
deci \(d\) \(10^{-1}\)
centi \(c\) \(10^{-2}\)
milli \(m\) \(10^{-3}\)
micro \(\mu\) \(10^{-6}\)
nano \(n\) \(10^{-9}\)
pico \(p\) \(10^{-12}\)

 

Self-Check Quiz

1. Convert \(7.2 \times 10^{-7}\) F into engineering and prefix notation.

2. Which of the following are equivalent?

  • \(1.5 \times 10^9 Hz\)
  • \(1500 \times 10^6 Hz\)
  • \(1.5 GHz\)

3. Explain one situation where scientific notation communicates more clearly than prefix notation.


Designed by Matthew Cheung. This work is licensed under a Creative Commons Attribution 4.0 International License.

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