Basic Mathematics | Chapter 1

1.1: The Integers

Lecture

Notes

Positive integers:   1,2,3,4,…1, 2, 3, 4, \dots

Natural Numbers:   0,1,2,3,…0, 1, 2, 3, \dots

Origin of the number line is zero.

Negative integers:   −1,−2,−3,−4,…-1, -2, -3, -4, \dots

Integers:   …,−3,−2,−1,0,1,2,3,…\dots, -3, -2, -1, 0, 1, 2, 3, \dots

N1.   0+a=a+0=a0 + a = a + 0 = a

N2.   a+(−a)=0and also−a+a=0a + (-a) = 0 \qquad \text{and also} \qquad -a + a = 0

−a-a is the additive inverse of aa

1.2. Rules for Addition

Lecture

Notes

Commutativity.   a+b=b+aa + b = b + a

Associativity.   (a+b)+c=a+(b+c)(a + b) + c = a + (b + c)

N3.   If a+b=0, then b=−a and a=−b\text{If } a + b = 0\text{, then } b = -a \text{ and } a = -b

N4.   a=−(−a)a = -(-a)

N5.   −(a+b)=−a−b-(a + b) = -a - b

1.3. Rules for Multiplication

Commutativity.   ab=baab = ba

Associativity.   (ab)c=a(bc)(ab)c = a(bc)

N6.   1a=aand0a=01a = a \qquad \text{and} \qquad 0a = 0

Distributivity.   a(b+c)=ab+acand(b+c)a=ba+caa(b + c) = ab + ac \qquad \text{and} \qquad (b+c)a = ba + ca

N7.   (−1)a=−a(-1)a = -a

N8.   −(ab)=(−a)b-(ab) = (-a)b

N9.   −(ab)=a(−b)-(ab) = a(-b)

N10.   (−a)(−b)=ab(-a)(-b) = ab

Exponents exist:   an=a1∗a2∗a3∗...∗ana^n = a_1 * a_2 * a_3 * ... * a_n

N11.   am+n=amana^{m+n} = a^ma^n

N12.   (am)n=amn(a^m)^n = a^{mn}

These formulas are used a lot, and should be remembered:

(a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2 (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2 (a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2

1.4. Even and Odd Integers: Divisibility