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IB Mathematics AI SL Operations with numbers Study Notes - New Syllabus

IB Mathematics AI SL Operations with numbers Study Notes

LEARNING OBJECTIVE

  • Operations with numbers in the form a×10k where 1≤a<10 and k is an integer.

Key Concepts: 

  • Standard Form

MAI HL and SL Notes – All topics

Standard Form 

Standard Form (also known as Scientific Notation) is a way of writing very large or very small numbers in a compact and manageable format.

A number is in standard form when it is written as:

$
a \times 10^n
$

Where:

$1 \leq a < 10$
$n$ is an integer (positive or negative)

This format makes calculations easier and helps when dealing with measurements in physics, astronomy, or computing.

Example

Converting 350,000 to Standard Form

Use this table to give in standard form.

▶️Answer/Explanation

Solution:

$
350000 = 3 \times 10^5 + 5 \times 10^4 + 0 \times 10^3 + \dots
$

But for standard form, 

$
a \times 10^n \quad \text{where } 1 \leq a < 10
$

$
350000 = 3.5 \times 10^5
$

The decimal point moved 5 places to the left
So the power of 10 is positive 5

Important Note on Notation

When writing numbers in standard form for exams or formal mathematics:

Calculator or computer notation is not acceptable.

$\text{5.2E30 is incorrect}$

 It should be written as:

$\rm{5.2 \times 10^{30}}$

Always use the full mathematical notation with the multiplication sign and power of 10 written as an exponent.

With or without GDC

Evaluate:

$
(4.5 \times 10^6) \times (2 \times 10^3)
$

Show Both methods properly.

▶️Answer/Explanation

Solution:

Method 1: Without GDC (Manual Calculation)

Step 1: Multiply the numbers:

$
4.5 \times 2 = 9
$

Step 2: Apply the laws of indices:

$
10^6 \times 10^3 = 10^{6+3} = 10^9
$

Step 3: Combine:

$
9 \times 10^9
$

Final Answer:

$
\rm{9 \times 10^9}
$

Method 2: With GDC (Calculator Use)

Input on GDC:

(4.5 × 10^6) × (2 × 10^3)

Calculator Output:

$\text{9E9}$

 Do not write this as 9E9 in exams.
 Instead, convert it to proper standard form:

Final Answer

$
\rm{9 \times 10^9}
$

Structure of Standard Form

The standard form has two components:

A number $a$ between 1 and 10
A power of 10 (i.e., $10^n$) indicating how many places the decimal point has moved

Converting Numbers to Standard Form

 A. Large Numbers

Steps:

1. Place the decimal point after the first non-zero digit.
2. Count how many places the decimal has moved to the left.
3. Use a positive power of 10.

Convert 72,000,000 to standard form.

$
72,000,000 = 7.2 \times 10^7
$

Example

Convert 5,420,000 to Standard Form

Show all the steps you have perform.

Answer must be in standard form.

▶️Answer/Explanation

Solution:

The number must lie between $1 \leq x < 10$,
so the number will begin as:

$
5.42
$

Maintain the value of the number by multiplying that decimal by a power of 10:

$
5.42 \times 1,000,000 = 5,420,000
$

 Write the power of ten as an exponent:

$
1,000,000 = 10^6
$

 Final answer in standard form:

$
\rm{5.42 \times 10^6}
$

 B. Small Numbers

Steps:

1. Move the decimal point to the right until you have one non-zero digit before it.
2. Count how many places it moved.
3. Use a negative power of 10.

Convert 0.00046 to standard form.

$
0.00046 = 4.6 \times 10^{-4}
$

Example

Convert 0.00081 to Standard Form

Show all the steps you have perform.

Answer must be in standard form.

▶️Answer/Explanation

Solution:

Identify the non-zero digits and rewrite them as a decimal number between 1 and 10:

$
0.00081 = 8.1 \times \text{(some power of 10)}
$

Determine what power of 10 is needed to preserve place value:

$
8.1 \div 10,000 = 0.00081
$

Express the power of ten as an exponent:

$
\frac{1}{10,000} = 10^{-4}
$

Final answer in standard form:

$
\rm{8.1 \times 10^{-4}}
$

Operations with Numbers in Standard Form

Example

Given:

 $A = 3.2 \times 10^5$
 $B = 4.5 \times 10^3$

Perform All the Mathematical operation $(+/-/×/÷)$ on $A$ and $B$. Answer should be standard form.

▶️Answer/Explanation

Solution:

1. Multiplication

$
(3.2 \times 10^5) \times (4.5 \times 10^3)
$

Multiply the base numbers:

$
3.2 \times 4.5 = 14.4
$

Add the exponents of 10:

$
10^5 \times 10^3 = 10^8
$

Combine:

$
14.4 \times 10^8 = 1.44 \times 10^9
$

$
\rm{1.44 \times 10^9}
$

2. Division

$
\frac{3.2 \times 10^5}{4.5 \times 10^3}
$

Divide the base numbers:

$
\frac{3.2}{4.5} \approx 0.7111
$

Subtract the exponents:

$
10^5 \div 10^3 = 10^{2}
$

Combine:

$
0.7111 \times 10^2 = 7.111 \times 10^1
$

$
\rm{7.111 \times 10^1}
$

3. Addition

$
(3.2 \times 10^5) + (4.5 \times 10^3)
$

Convert to ordinary numbers:

$
3.2 \times 10^5 = 320000 \quad \text{and} \quad 4.5 \times 10^3 = 4500
$

Add:

$
320000 + 4500 = 324500
$

Convert back to standard form:

$
324500 = 3.245 \times 10^5
$

$
\rm{3.245 \times 10^5}
$

4. Subtraction

$(3.2 \times 10^5) – (4.5 \times 10^3)$

Convert to ordinary numbers:

$320000 – 4500 = 315500$

Convert back to standard form:

$315500 = 3.155 \times 10^5$

$\rm{3.155 \times 10^5}$

Real-Life Examples of Standard Form

Standard form is used in science, engineering, and astronomy:

Astronomical Distances

The distance from the Sun to Mars:

141,700,000 miles = $\mathbf{1.417 \times 10^8}$ miles
228,000,000 km = $\mathbf{2.28 \times 10^8}$ km

Atomic Masses

Mass of a proton or neutron = $\mathbf{1.67 \times 10^{-27}}$ kg
Mass of an electron = $\mathbf{9.11 \times 10^{-31}}$ kg

Other Use Cases:

Size of bacteria: $1 \times 10^{-6}$ m
Computer microchip transistors: $2 \times 10^{-9}$ m
World population: $8.1 \times 10^9$ people

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