Integer (整数)

Abstract

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• Discrete objects, for example there is a clear separation between $1$ and $2$, unlike there can be $1.25$ between $1.2$ and $1.3$
• Used a lot in Discrete Math
• The mathematical symbol is $\mathbb{Z}$ (315, -9045), positive integers are represented with $\mathbb{N}$ (0,1,2,3,4…)

Important

$0$ is neither negative or positive.

Integer Mathematical Model

• any integer can be obtain from ${\sum }_{i=0}^{i=n-1}{d}_i\times 10^i$
  • $n$: total number of digits in the given integer
  • $i$: the index position of the given integer counting from the right hand side
  • $d_i$: the value of the digit at position $i$

Given the integer $9213$, $n$ is $4$

${\sum }_{i=0}^{i=3}{d}_i\times 10^i=d_0\times 10^0+d_1\times 10^1+d_2\times 10^2+d_3\times 10^3$

Substitute in the digits, we get $3\times 10^0+1\times 10^1+2\times 10^2+9\times 10^3$

Simplify it, we obtain $3\times 1+1\times 10+2\times 100+9\times 1000=3+10+200+9000$

Finally we obtain $9213$

Theorems

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Theorem 4.4.1

• The Quotient-Remainder Theorem
• Given any Integer (整数) n and positive integer d, there exist unique integers q and r such that

$$(n=dq+r)\wedge (0\le r<d)$$

Theorem 4.6.1

• There is no greatest integer
• Proved by Proof by Contradiction (矛盾证明法)