October 27, 2025
Kuina-chan
In Episode 3 of Kuina-chan Mathematics, I explain integers, which include negative numbers in addition to natural numbers! This episode assumes you’ve read the series in order, beginning with Episode 1.
In Episode 2, I used axioms of sets, natural numbers, and addition to prove that “1+1=2”. But even without those axioms, we are confident that “1+1=2” is true. So instead of continuing to prove things like “1\times1=1” or “1-1=0” using the same method, I will assume these are provable if needed, and from now on focus on things that we really don’t know if they are true or not.

1.Integers

In Episode 2, I represented natural numbers as the set “\mathbb{N}=\{0,1,2,3,4,\dots\}”. If we include numbers with minus signs (except for 0), we get the set of integers. So the set of all integers is “\mathbb{Z}=\{\dots-3,-2,-1,0,1,2,3,\dots\}”.
Numbers greater than 0 are called positive, and numbers less than 0 are called negative. 0 is neither positive nor negative.
You probably know that for any two integers a and b, you can do addition “a+b”, subtraction “a-b”, and multiplication “a\timesb”. Multiplication “a\timesb” is also written as “a\cdotb”, and often the multiplication sign is omitted and written as “ab”. I will use that style from now on.

1.1Exponentiation


For an integer a and a non-negative integer b, the number you get by multiplying a b times is called an exponentiation and written as “a^{b}”. For example, “2^{3}” means “2\cdot2\cdot2”, which is 8. “10^{4}” means “10\cdot10\cdot10\cdot10”, which is 10000.
Also, for any non-zero number a, we define “a^{0}=1”. For example, “2^{0}=1” and “(-5)^{0}=1”.
Note

If you look at “2⁵ = 32”, “2⁴ = 16”, “2³ = 8”, “2² = 4”, “2¹ = 2”, you can see the result is halved each time. So it’s natural to think that “2⁰ = 1”.

“0^{0}” is sometimes defined as “1” for convenience, but for various reasons, it is often not defined.
Note

One reason why “0⁰” is usually not defined is that if you look at “3⁰ = 1”, “2⁰ = 1”, “1⁰ = 1”, it seems natural to think “0⁰ = 1”. But if you look at “0³ = 0”, “0² = 0”, “0¹ = 0”, it seems natural to think “0⁰ = 0”. These two ideas conflict.

Exponentiation follows the rules below:
Exponent Rules
  1. a^{m}\cdota^{n}=a^{m}^{+}^{n}
  2. a^{m}/a^{n}=a^{m}^{-}^{n}
  3. (a^{m})^{n}=a^{m}^{\cdot}^{n}
  4. (a\cdotb)^{n}=a^{n}\cdotb^{n}
(1) is clear because multiplying “a m times” and “a n times” gives “a m+n times”.
(2) is subtraction because division reduces the number of as.
(3) means you have “a m times” repeated n times, so you get “a m\cdotn times”.
(4) means you have “a\cdotb” repeated n times, which becomes “a n times and b n times”.

1.2Absolute Value


The distance from 0 of an integer a is called its absolute value, written as “|a|”. For example, “|5|=5” and “|-3|=3”.
You can think of absolute value as “if it’s positive, keep it as is; if it’s negative, remove the minus sign”.
More strictly, it’s defined as:
Definition of Absolute Value

|a| is defined as:

  • If a\geq0, then |a|=a.
  • If a<0, then |a|=-a.
For example, if a=-3, then a<0, so “|a|=-a=-(-3)=3”.

2.Properties of Integers

Now I will explain various properties of integers.

2.1Quotient and Remainder


When dividing two integers (a/b), the result is not always an integer. So we define the quotient and remainder to make the result an integer.
The quotient of “a/b” is the number of items each person gets when a items are distributed among b people. The remainder is the number of items left over. For example, “7/3” gives a quotient of 2 and a remainder of 1.
Saying “7/3 has quotient 2 and remainder 1” means “distributing 7 items among 3 people gives 2 items each and 1 left over”. In other words, “2\cdot3+1=7”. So we define the quotient q and remainder r of a/b as the numbers that satisfy “q\cdotb+r=a”.
Definition of Quotient and Remainder

For integers a and non-zero b, the quotient and remainder of a/b are the integers q and r that satisfy: q\cdotb+r=a and 0\leqr<b.

For example, “8/5” gives q=1 and r=3, and “1\cdot5+3=8” with “0\leq3<5”.
If b=0, the quotient and remainder are not defined. So “3/0” is undefined.

2.2Divisible, Divisor, Multiple


If the remainder of a/b is 0, we say that b divides a. For example, “4/2” has remainder 0, so 2 divides 4. “12/3” also has remainder 0, so 3 divides 12.
If b divides a, then b is a divisor of a, and a is a multiple of b. For example, 2 is a divisor of 4, and 4 is a multiple of 2.
The divisors of 12 are all the numbers that divide 12, so in order: “-12,-6,-4,-3,-2,-1,1,2,3,4,6,12”. The multiples of 2 are “\dots-6,-4,-2,0,2,4,6,\dots”, which are all even numbers.
1 and -1 divide all integers, so their multiples are all integers. All integers except 0 divide 0, so the divisors of 0 are all integers except 0.

2.3Common Divisors and Common Multiples


Now let’s look at common divisors and multiples of two or more integers.
A common divisor of a and b is a number c that divides both a and b. For example, 2 divides both 4 and 6, so 2 is a common divisor of 4 and 6.
A common multiple of a and b is a number c that is divisible by both a and b. For example, 10 is divisible by both 2 and 5, so 10 is a common multiple of 2 and 5.
You can also define common divisors and multiples for three or more numbers.

2.4Greatest Common Divisor and Least Common Multiple


The greatest common divisor of a and b is the largest of their common divisors, written as “\rm{g}\rm{c}\rm{d}(a,b)”. The least common multiple of a and b is the smallest of their positive common multiples, written as “\rm{l}\rm{c}\rm{m}(a,b)”.
Note

“gcd” stands for “greatest common divisor”, and “lcm” stands for “least common multiple”.

For example, the divisors of 8 are “-8,-4,-2,-1,1,2,4,8”, and the divisors of 6 are “-6,-3,-2,-1,1,2,3,6”. Their common divisors are “-2,-1,1,2”, so “\rm{g}\rm{c}\rm{d}(8,6)=2”.
The positive multiples of 8 are “8,16,24,32,\dots”, and of 6 are “6,12,18,24,30,\dots”. Their common multiples are “24,48,72,\dots”, so “\rm{l}\rm{c}\rm{m}(8,6)=24”.
For positive integers a and b, the rule “a\cdotb=\rm{g}\rm{c}\rm{d}(a,b)\cdot\rm{l}\rm{c}\rm{m}(a,b)” holds. For example, “8\cdot6=2\cdot24” gives “48=48”. So if you know one of gcd or lcm, you can easily calculate the other.

2.5Euclidean Algorithm


Finding the greatest common divisor by listing all divisors takes time, but the Euclidean Algorithm is a fast method.
Euclidean Algorithm
  1. Let a be the larger and b the smaller of the two integers.
  2. Let r be the remainder of a/b.
  3. If r\neq0, then \rm{g}\rm{c}\rm{d}(a,b)=\rm{g}\rm{c}\rm{d}(b,r). Repeat from step 1.
  4. If r=0, then \rm{g}\rm{c}\rm{d}(a,b)=b. (Done)
For example, to find the gcd of 128 and 80:
GCD of 128 and 80
  • a=128, b=80.
  • 128/80 gives remainder r=48, so \rm{g}\rm{c}\rm{d}(128,80)=\rm{g}\rm{c}\rm{d}(80,48).
  • \rm{g}\rm{c}\rm{d}(80,48)=\rm{g}\rm{c}\rm{d}(48,32)=\rm{g}\rm{c}\rm{d}(32,16).
  • 32/16 gives remainder r=0, so \rm{g}\rm{c}\rm{d}(32,16)=16.
  • Therefore, \rm{g}\rm{c}\rm{d}(128,80)=16.
In general, repeating division is easier than listing divisors, so this method is useful.

3.Prime Numbers

A number p\geq2 is called a prime number if its only positive divisors are 1 and p. For example, 5 is prime because its only positive divisors are 1 and 5. 4 is not prime because it also has 2 as a divisor.
In other words, a prime number is a number \geq2 that cannot be divided by any positive integer other than 1 and itself. Numbers that are not prime are called composite numbers.
The prime numbers in order are “2,3,5,7,11,13,17,19,23,29,\dots”. There are infinitely many primes. Their pattern seems irregular, and people have studied it from ancient times to today.
You can find prime numbers using the Sieve of Eratosthenes, which uses the idea that numbers not divisible by other primes are prime.
Sieve of Eratosthenes
Sieve of Eratosthenes

3.1Prime Factorization


Every positive integer can be written as a product of prime numbers. For example, “4=2\cdot2”, “6=2\cdot3”, “50=2\cdot5\cdot5”. This is called prime factorization.
Each prime number in the factorization is called a prime factor. For example, “10=2\cdot5”, so the prime factors of 10 are 2 and 5.
Every positive integer can be uniquely factorized into primes (ignoring the order). For example: “1=2^{0}3^{0}5^{0}\dots” “2=2^{1}3^{0}5^{0}\dots” “3=2^{0}3^{1}5^{0}\dots” “4=2^{2}3^{0}5^{0}\dots” “5=2^{0}3^{0}5^{1}\dots” “6=2^{1}3^{1}5^{0}\dots”. This property is called the uniqueness of prime factorization, and it’s useful for proving other theorems.
We don’t include “1” as a prime number because if we did, then: “2=1^{0}2^{1}\dots=1^{1}2^{1}\dots=1^{2}2^{1}\dots=1^{3}2^{1}\dots” This would break the uniqueness of prime factorization.

3.2Relatively Prime


Two integers a and b are relatively prime if they have no common divisors other than 1 and -1, that is, “\rm{g}\rm{c}\rm{d}(a,b)=1”. For example, “\rm{g}\rm{c}\rm{d}(9,20)=1”, so 9 and 20 are relatively prime.
For positive integers a and b, being relatively prime means they have no common prime factors. For example, “9=3\cdot3” and “20=2\cdot2\cdot5” have no common prime factors, so they are relatively prime.

4.Congruence

Back to remainders: “5 divided by 3 has remainder 2”, and “8 divided by 3 also has remainder 2”. So in the world of remainders modulo 3, “5=8”. When two numbers have the same remainder when divided by m, we say they are congruent modulo m, and write “5\equiv8 (\rm{m}\rm{o}\rm{d} 3)”.
In general, if a and b have the same remainder when divided by m, we write “a\equivb (\rm{m}\rm{o}\rm{d} m)”. If not, we write “a\not\equivb (\rm{m}\rm{o}\rm{d} m)”. These are called congruence equations.
For example, “3\equiv1 (\rm{m}\rm{o}\rm{d} 2)” because both have remainder 1. “5\not\equiv6 (\rm{m}\rm{o}\rm{d} 4)” because their remainders are different.
Congruence equations stay true if you add, subtract, or multiply both sides by the same number.
Properties of Congruence

If a\equivb (\rm{m}\rm{o}\rm{d} m), then for any integer n:

  1. a+n\equivb+n (\rm{m}\rm{o}\rm{d} m)
  2. a-n\equivb-n (\rm{m}\rm{o}\rm{d} m)
  3. a\cdotn\equivb\cdotn (\rm{m}\rm{o}\rm{d} m)
For example, “5\equiv8 (\rm{m}\rm{o}\rm{d} 3)” implies “500\equiv800 (\rm{m}\rm{o}\rm{d} 3)”.

5.Indeterminate Equation

Finally, let’s try a real problem using the properties of integers. It’s called an indeterminate equation.
An equation is a problem like “4x=8” where you find values of variables that make the equation true. These values are called solutions.
An indeterminate equation is one that has infinitely many solutions. For example, “x+2y=3” has solutions like “x=3,y=0” and “x=1,y=1”.
Sometimes, adding conditions makes the number of solutions finite. Let’s solve a puzzle-like problem using that idea.

5.1Problem


Here’s an indeterminate equation problem:
Indeterminate Equation Problem
Problem

Let “1234” become “4321” by reversing the digits. A 4-digit number N becomes 4 times itself when reversed. Find the value of N.

5.2Solution


First, let’s build the equation. Let the digits of N be a, b, c, d from left to right. For example, if N=1234, then a=1, b=2, c=3, d=4. Then N=1000a+100b+10c+d.
Reversing N gives 1000d+100c+10b+a, and since it’s 4 times the original, we get:
Indeterminate Equation

4\cdot(1000a+100b+10c+d)=1000d+100c+10b+a

This equation has 4 variables and many solutions, so let’s use conditions to narrow it down.

5.3Finding a


If a=0, N is not 4 digits, so a>0. If a\geq3, then 4N is 5 digits, so a<3. So a is either 1 or 2.
If a=1, then the equation becomes “4\cdot(1000+100b+10c+d)=1000d+100c+10b+1”. The rightmost digit is 1, but multiplying by 4 never gives a number ending in 1. So no solution. Therefore, a=2.

5.4Finding d


Substitute a=2 into the equation: “4\cdot(2000+100b+10c+d)=1000d+100c+10b+2”. The rightmost digit is 2, so d must be a digit such that 4\cdot(\dots) ends in 2. Only 3\cdot4=12 and 8\cdot4=32 work, so d=3 or 8.
Try d=3: equation becomes “4\cdot(2000+100b+10c+3)=3000+100c+10b+2”. Solving gives “b=(6c-501)/39”, which is negative for all c=0 to 9. So d\neq3, and d=8.

5.5Finding b and c


Substitute d=8: equation becomes “4\cdot(2000+100b+10c+8)=8000+100c+10b+2”. Solving gives “b=(2c-1)/13”. Only c=7 makes this an integer, so c=7.
Then b=(2\cdot7-1)/13=13/13=1, so b=1.
So a=2, b=1, c=7, d=8, and N=2178. Check: 2178\cdot4=8712, which is the reverse of 2178.
This time, I introduced basic properties of integers. Next time, I will explain real numbers, and important concepts like functions and mappings!
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