IB Mathematics AA 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
- IBDP Maths AA SL- IB Style Practice Questions with Answer-Topic Wise-Paper 1
- IBDP Maths AA SL- IB Style Practice Questions with Answer-Topic Wise-Paper 2
- IB DP Maths AA HL- IB Style Practice Questions with Answer-Topic Wise-Paper 1
- IB DP Maths AA HL- IB Style Practice Questions with Answer-Topic Wise-Paper 2
- IB DP Maths AA HL- IB Style Practice Questions with Answer-Topic Wise-Paper 3
Operations with Numbers in the Form \( a \times 10^k \)
Operations with Numbers in the Form \( a \times 10^k \)
Scientific Notation: A number written in the form \( a \times 10^k \), where:
- \( 1 \leq a < 10 \) – the coefficient is a number between 1 (inclusive) and 10 (exclusive).
- \( k \in \mathbb{Z} \) – the exponent is an integer, positive or negative.
Addition & Subtraction
To add or subtract, rewrite the numbers with the same power of 10.
Steps:
- Convert both numbers to have the same exponent.
- Add or subtract the coefficients.
- Convert the result back to proper scientific form if needed.
Example: \( 3.2 \times 10^4 + 4.5 \times 10^3 = 3.2 \times 10^4 + 0.45 \times 10^4 = 3.65 \times 10^4 \)
Multiplication
Multiply the coefficients and add the exponents.
Rule: \( (a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n} \)
Adjust the result to be in proper form if necessary.
Example: \( (2.5 \times 10^3) \times (4 \times 10^5) = 10 \times 10^8 = 1.0 \times 10^9 \)
Division
Divide the coefficients and subtract the exponents.
Rule: \( \dfrac{a \times 10^m}{b \times 10^n} = \left( \dfrac{a}{b} \right) \times 10^{m-n} \)
Adjust the result to proper form if necessary.
Example: \( \dfrac{6.0 \times 10^6}{3.0 \times 10^2} = 2.0 \times 10^4 \)
Conversion to and from Decimal
- If \( k > 0 \): move the decimal point right \( k \) times.
- If \( k < 0 \): move the decimal point left \( |k| \) times.
Example: \( 3.4 \times 10^3 = 3400 \), and \( 5.6 \times 10^{-2} = 0.056 \)
Example
Evaluate \( (2.5 \times 10^3) + (3.7 \times 10^2) \)
▶️ Answer/Explanation
Rewrite \( 2.5 \times 10^3 = 2500 \) and \( 3.7 \times 10^2 = 370 \)
Now add: \( 2500 + 370 = 2870 \)
Convert back to standard form: \( 2870 = 2.87 \times 10^3 \)
Final Answer: \( \boxed{2.87 \times 10^3} \)
Example
Evaluate \( (5.0 \times 10^{-5}) – (2.0 \times 10^{-6}) \)
▶️ Answer/Explanation
Convert both to standard decimal:
\( 5.0 \times 10^{-5} = 0.00005 \), \( 2.0 \times 10^{-6} = 0.000002 \)
Subtract: \( 0.00005 – 0.000002 = 0.000048 \)
Convert back: \( 0.000048 = 4.8 \times 10^{-5} \)
Final Answer: \( \boxed{4.8 \times 10^{-5}} \)
Very Large and Very Small Numbers
Very Large and Very Small Numbers
Scientific notation is essential for expressing quantities that are extremely large or extremely small in a compact and understandable form.
Astronomical Distances (Very Large)
- Distance from Earth to Sun: \( 1.496 \times 10^8 \) km
- Diameter of the Milky Way Galaxy: \( 1.0 \times 10^5 \) light-years ≈ \( 9.46 \times 10^{20} \) m
- Observable Universe: ≈ \( 8.8 \times 10^{26} \) m
Sub-Atomic Particles (Very Small)
- Radius of a hydrogen atom: ≈ \( 5.3 \times 10^{-11} \) m
- Mass of an electron: \( 9.11 \times 10^{-31} \) kg
- Charge of an electron: \( 1.6 \times 10^{-19} \) C
Global Financial Figures (Very Large)
- World GDP (2024): ≈ \( 1.06 \times 10^{14} \) USD
- National Debt of the USA: ≈ \( 3.3 \times 10^{13} \) USD
- Market Capitalization of Apple Inc.: ≈ \( 3 \times 10^{12} \) USD
These quantities are too complex to write or interpret easily in standard decimal form. Scientific notation simplifies reading, writing, and calculating with such numbers in science, engineering, and finance.
Example
Evaluate \( (3 \times 10^8) \times (4 \times 10^5) \)
▶️ Answer/Explanation
This is multiplication of numbers in standard form:
\( (3 \times 4) \times (10^8 \times 10^5) = 12 \times 10^{13} \)
Convert to standard form: \( 12 \times 10^{13} = 1.2 \times 10^{14} \)
Final Answer: \( \boxed{1.2 \times 10^{14}} \)
Example
Evaluate \( \frac{6 \times 10^{-6}}{2 \times 10^{-2}} \)
▶️ Answer/Explanation
Apply division rule: \( \frac{6}{2} \times 10^{-6 – (-2)} = 3 \times 10^{-4} \)
Final Answer: \( \boxed{3 \times 10^{-4}} \)
Graphic Display Calculator TI-84 Plus
Graphic Display Calculator TI-84 Plus
To write a number in standard form on the TI-84 Plus, use the EE key. To convert a number to standard form, use the SCI mode.
Scientific notation expresses numbers as a product of a decimal number \( a \) and a power of ten \( 10^k \), where:
- \( 1 \leq a < 10 \)
- \( k \in \mathbb{Z} \) (an integer)
- Example: \( 5.2 \times 10^{30} \) is correct; writing
5.2E30is not acceptable.
Note: Calculator or computer notation such as 3.0E8, 6.02e23, or 1e-9 should never be used in formal written work. Always express numbers in the form \( a \times 10^k \).
Example
Use the TI-84 Plus Graphic Display Calculator to express 1,230,000,000 in standard form.
▶️ Answer/Explanation
Step-by-step using the TI-84 Plus:
- Press
[MODE]and scroll down to select SCI (for scientific notation). Press[ENTER]. - Return to the home screen by pressing
[2nd] → [MODE](Quit). - Now type
1230000000and press[ENTER].
The GDC will display:
1.23E9
Final Answer (in correct written form):
\( \rm{1.23 \times 10^9} \)
Note: You must write the answer as \( 1.23 \times 10^9 \). The calculator output 1.23E9 is not acceptable in exams or formal written work.
Use of Scientific Notation in the Sciences
Use of Scientific Notation in the Sciences
Scientific notation and understanding orders of magnitude are crucial for comparing, analyzing, and communicating measurements in the sciences. Below are applications across disciplines:
Chemistry: Avogadro’s Number
- Avogadro’s number: \( N_A = 6.022 \times 10^{23} \) mol⁻¹ – the number of atoms, ions, or molecules in one mole of a substance.
- Used in mole-to-particle conversions, stoichiometry, and gas law calculations.
- Example: One mole of H₂O contains \( 6.022 \times 10^{23} \) water molecules.
Physics: Order of Magnitude
- An order of magnitude refers to the power of ten when expressing a number in scientific notation.
- Used to estimate and compare physical quantities without exact calculations.
- Example: The speed of light \( (3 \times 10^8 \text{ m/s}) \) is 8 orders of magnitude greater than walking speed \( (\approx 1 \text{ m/s}) \).
Biology: Microscopic Measurements
- Cellular and molecular sizes are typically expressed in micrometers (µm = \( 10^{-6} \) m) or nanometers (nm = \( 10^{-9} \) m).
- Example: Diameter of a red blood cell ≈ \( 7.5 \times 10^{-6} \) m.
- DNA helix diameter ≈ \( 2 \times 10^{-9} \) m.
Sciences Group Subjects: Uncertainty and Precision
- Uncertainty: Reflects the doubt in a measurement; often stated as ± value.
- Precision: The degree of reproducibility or consistency of a set of measurements.
- Scientific notation helps present both values clearly, especially when dealing with small/large uncertainties.
- Example: Mass = \( 2.36 \times 10^{-3} \pm 0.01 \times 10^{-3} \) kg
Scientific notation is not only a way to simplify large/small values, but also a powerful tool for expressing accuracy, comparing magnitudes, and applying mathematical reasoning in scientific research.
Example
The distance from Earth to the Sun is \( 1.496 \times 10^8 \) km. Light travels at \( 3.0 \times 10^5 \) km/s. How long does light take to reach Earth from the Sun?
▶️ Answer/Explanation
Use: time = distance ÷ speed
\( \frac{1.496 \times 10^8}{3.0 \times 10^5} = \frac{1.496}{3.0} \times 10^{8 – 5} \)
\( = 0.4987 \times 10^3 = 4.987 \times 10^2 \) seconds
Final Answer: \( \boxed{498.7 \text{ seconds}} \)
