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[h] IB Mathematics AA HL Flashcards- Standard Form
[q] Standard form
A number in the format , \( \pm a \times10^k \) , where \(1\leqslant a\leqslant 10 \) and \(k\in Z\) (k is an integer)
Number in the form \(a \times 10^ k\) such as \(4.7 \times 10^5\) instead of 475,000 may be called Specific notation or Standard form or not given any specific name.
[q]
CONDITIONS // The number 470,000 could theoretically be written: \( 4.7 \times 10^5 \), \( 47 \times 10^4 \), \( 0.47 \times 10^6 \), and so on. However, to simplify things, we say that ‘a’ & ‘k’ must have the following:
– \( 1 \leq |a| < 10 \)
– \( k \in \mathbb{Z} \)
Meaning: ‘a’ is between 1 and 10 (+ve or -ve), and ‘k’ is an integer.
[a]
E.G.1 // \( 5 \times 10^{3.2} \): NOT in the correct form (k is not an integer)
E.G.2 // \( 0.89 \times 10^{-4} \): NOT, because \( a < 1 \). Should be: \( 8.9 \times 10^{-5} \)
E.G.3 // \( 2.3 \times 10^8 \): ✓, this means 230,000,000
[q]
GENERAL USAGE // This format of writing numbers is frequently used in science, where we want to save time by avoiding writing very large or very small numbers in full. For example, the width of an atom is ~\( 3 \times 10^{-10} \) m.
[a]
I.B. MATH USAGE // It is extremely unlikely that you will ever see a standalone question about this. Far more likely is to see this used as a small element of a question on some other topic.
[q]
E.G.4 // Find \( x \) (in the form \( a \times 10^k \), \( 1 \leq a < 10 \), \( k \in \mathbb{Z} \))
Using sine rule:
\[
\frac{x}{\sin 21^\circ} = \frac{1.6 \times 10^7}{\sin 57^\circ}
\]
Solving for \( x \):
\[
x = 0.834 \times 10^7
\]
Change to correct form:
\[
x = 8.34 \times 10^6
\]
[q] Notes on GDC
[a]
[q] Standard Form – Solved Examples 1
The diameter of a spherical planet is 6 × 104 km .
(a) Write down the radius of the planet. [1]
The volume of the planet can be expressed in the form π(a × 10k) km3 where 1 ≤ a <10 and k ∈ \(\mathbb{Z}\)
(b) Find the value of a and the value of k .
[a] Answer – Solved Examples 1
(a) \(3 \times 10^4\)
(b) \(\frac{4}{3}\pi(3 \times 10^4)^3\)
\(=\frac{4}{3}\pi\times 27 \times 10^{12}\)
\(=\pi(3.6 \times 10^{13})(km)^3\)
Hence
\(a= 3.6 \) , \(k =13\)
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