Basic manipulations: additions, multiplications and powers

[Jack]

When learning the numbers, we start with counting, and then introduce additions, multiplications and powers.
- additions: continued counting. Example: \(3 + 4 = 7\) and \( 3 + 4 + 5 = 12 \);
- multiplications: repeated additions. Example: \(3 + 3 + 3 + 3 + 3 = 5 \cdot 3 \);
- powers: repeated multiplications. Example: \( 3 \cdot 3 \cdot 3 \cdot 3= 3^4 \).


For a given number \(n\), its opposite number \(\hat n\) defines a number with the property that \(n + \hat n =0\). It is common to write \(\hat{n}=-n\).
Subtraction is another manipulation defined as the addition of the opposite number: \(m - n = m + \hat{n}.\)

For a given nonzero number \(n\), its reciprocal number \(\tilde{n}\) defines a number with the property that \(n \cdot \tilde{n} =1\). It is common to write \(\tilde{n} = \frac{1}{n}\).
Division is another manipulation defined as the multiplication of the reciprocal number: \(m \div n = m \cdot \tilde{n} = m \cdot \frac{1}{n}.\)

The opposite manipulation of powers is the roots. For example,
- \( 3^2 = 9 \) defines the (square) root \(9^{1/2} = 3\);
- \( 3^3 = 27 \) defines the cubic root \(27^{1/3} = 3\);
- \( 3^4 = 81 \) defines the 4-th root \(27^{1/4} = 3\).

Laws of manipulations.

Additions and Multiplications:
- Commutative Laws: \(a + b = b + a, a\cdot b = b\cdot a\);
- Associative Laws: \( (a + b) + c = a + (b + c), (a\cdot b) \cdot c = a \cdot ( b\cdot c) \);
- Distributive Law: \( (a + b) \cdot c = a \cdot c + b\cdot c \).

Laws of Powers: let \(a > 0, b > 0\) be two positive numbers, \(m, n\) be two integers. Then
- Products of Powers: \(a^{m} \cdot a^{n} = a^{m + n}\);
- Powers of Powers: \( (a^{m})^{n} = a^{m \cdot n}\);
- Powers of Products: \( (a \cdot b)^{n} = a^n \cdot b^n \).

Note that one has to be very careful when generalizing the powers laws to negative bases and fractional powers. We will talk more about this in the next post.