Negative Exponents:
Negative exponents can be written using the below rule
\(n^{-a} = \frac{1}{n^{a}}\)
Example:
\(4^{-2} = \frac{1}{4^{2}}\)
Power of 10:
Power of 10 can be used to explain the larger values in expanded form and decimal values can be expressed using powers of 10.
Example:
1. \(47561\)
\(=(4 \times 10000) + (7 \times 1000) + (5 \times 100) + (6 \times 10) + 1 \times 1\)
\(=(4 \times 10^{4}) + (7 \times 10^{3}) + (5 \times 10^{2}) + (6 \times 10^{1}) + 1 \times 10^{0}\)
2. \(561.903\)
\(=(5 \times 100) + (6 \times 10) +(1 \times 1) + (9 \times \frac{1}{10}) + (0 \times \frac{1}{100}) + (3 \times \frac{1}{1000}\)
\(=(5 \times 10^{2}) + (6 \times 10^{1}) +(1 \times 10^{0}) + (9 \times 10^{-1}) + (0 \times 10^{-2}) + (3 \times 10^{-3}\)
Scientific Notation:
In scientific notation, we write numbers as \(x \times 10^{y}\), where \(x \geq1\) and \(x<10\) is the coefficient and y, is the exponent,is any integer.
Example:
1.\(3430000 = 3.43 \times 10^{6}\)
2. \(0.0059 = 5.9 \times 10^{-3}\)
Linear Vs Exponential Growth:
Linear Growth: Repeated addition of constant value
Exponential Growth: Repeated multiplication of constant value
Example:
To cover the distance between Earth and Moon
With linear growth it takes \(1,92,20,00,000\) steps with each step is 20 cm gain.
With exponential growth it takes \(46\) folds of a piece of paper.
Very Large Numbers & Order of Magnitude:
Scientific notation is used to express larger quantities.
Example:
Stars in Universe: \(2 \times 10^{23}\)
Ant population: \(2 \times 10^{16}\)
Time in Powers of Ten:
Modern and historical timelines expressed using power of \(10\)
Example:
\(10^{0}\) = \(1\) second
\(10^{1}\) = \(10\) second
\(10^{2}\) = \(1.6\) minutes
It goes on like this....
Historical Number Names (Jaina & Buddhist Texts)
Ancient systems named powers up to \(10^{96}\) and beyond that.
Examples:
Ayuta - \(10^{9}\)
Niyuta - \(10^{11}\)
Kankara - \(10^{13}\)