Standard Form – ( Scientific Notation )

Objective

• Understand the definition of standard form (scientific notation)
• Convert numbers between ordinary and standard form
• Perform operations: addition, subtraction, multiplication, and division in standard form

Definition

A number is written in standard form (or scientific notation) when it is expressed as: \(a \times 10^n\), where \(1 \leq a < 10\) and \(n \in \mathbb{Z}\)

• \(a\) is a number between 1 and 10 (not including 10)
• \(n\) is an integer indicating how many times the decimal point has moved

Examples (Conversion)

Convert ordinary numbers to standard form

(a) \(\quad 4500 = 4.5 \times 10^3\)

(b) \(\quad 0.0063 = 6.3 \times 10^{-3}\)

Convert standard form to ordinary numbers

(c) \(\quad 3.2 \times 10^4 = 32000\)

(d) \(\quad 9.8 \times 10^{-2} = 0.098\)

Binary Operations

Make the powers of 10 the same before adding or subtracting.

(e) Addition 

          \((3.2 \times 10^4) + (4.5 \times 10^3)\)

\(= (3.2 \times 10^4) + (0.45 \times 10^4)\)

\(= 3.65 \times 10^4\)

(f) Subtraction

         \((6.1 \times 10^{-2}) – (2.5 \times 10^{-3})\)

         \(= (6.1 \times 10^{-2}) – (0.25 \times 10^{-2})\)

         \(= 5.85 \times 10^{-2}\)

Multiplication in Standard Form
Multiply the numbers, then add the powers of 10.

(g) \((4 \times 10^3) \times (2 \times 10^5) = 8 \times 10^{8}\)

Division in Standard Form
Divide the numbers, then subtract the powers of 10.

(h)  \(\frac{6 \times 10^7}{2 \times 10^3} = 3 \times 10^4\)

Practice worksheet : Download

Question 1: Ordinary ➜ Scientific Notation

Convert the following ordinary numbers into scientific notation

a. \(0.00047\)

b. \(85000000\)

c. \(0.0062\)

d. \(9300\)

e. \(0.00000039\)

Question 2:  Scientific ➜ Ordinary Notation

Convert the following scientific notation numbers into ordinary (decimal) form

a. \(4.5 \times 10^6\)

b. \(8.3 \times 10^{-3}\)

c. \(9.9 \times 10^3\)

d. \(6.5 \times 10^{-4}\)

e. \(7.2 \times 10^{-6}\)

Question 3: Addition & Subtraction (Scientific Form)

Add or subtract the following using scientific notation

a. \(4.5 \times 10^3 + 3.2 \times 10^3\)

b. \(7.2 \times 10^5 + 8 \times 10^4\)

c. \(9.6 \times 10^6 – 3.2 \times 10^6\)

d. \(1.5 \times 10^3 – 1.2 \times 10^3\)

e. \(5.4 \times 10^4 + 3.6 \times 10^4\)

Question 4: Multiplication (Scientific Form)

Multiply the following in scientific notation

a. \((3 \times 10^4)(2 \times 10^5)\)

b. \((6 \times 10^2)(4 \times 10^3)\)

c. \((2.5 \times 10^1)(5 \times 10^2)\)

d. \((7.5 \times 10^3)(2 \times 10^2)\)

e. \((9 \times 10^4)(3 \times 10^5)\)

Question 5: Division (Scientific Form)

Divide the following in scientific notation

a. \(\frac{8 \times 10^6}{4 \times 10^2}\)

b. \(\frac{9 \times 10^3}{3 \times 10^1}\)

c. \(\frac{1.2 \times 10^5}{6 \times 10^2}\)

d. \(\frac{6 \times 10^7}{2 \times 10^3}\)

e. \(\frac{5.4 \times 10^8}{9 \times 10^4}\)