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Axioma

Mathematics · Foundational · 25 min · Chapter 1 of 28

Arithmetic and number sense

Numbers, the four operations, fractions, percent, factors, and the idea that every integer has a unique prime recipe.

The idea

Arithmetic is not a pile of facts to memorize. It is the study of quantity: how numbers combine, and which properties survive the combining. The commutative law says a + b = b + a and a × b = b × a. The associative law says grouping does not matter for a single operation. The distributive law is the one that actually mixes them: a(b + c) = ab + ac. Almost every later simplification is one of these three laws in disguise.

A fraction a/b means a pieces of size 1/b, with b not zero. Two fractions name the same number when their cross products agree: a/b = c/d exactly when ad = bc. Adding them needs a common size of piece, which is why the general rule uses the products ad and bc over the common denominator bd. You can always cancel a common factor afterward; cancelling before multiplying is the same fact, used earlier.

A prime is an integer greater than 1 whose only positive divisors are 1 and itself. The fundamental theorem of arithmetic says the prime factorization of an integer is unique. That uniqueness is why greatest common divisors and least common multiples can be read off from the smaller and larger powers of each prime. Euclid’s algorithm is faster: gcd(a, b) = gcd(b, a mod b), and it ends when the remainder is 0.

Hold onto this

  1. Place value is what makes a digit mean ten, a hundred, or a tenth.
  2. Fractions and decimals name the same quantities; percent is a fraction with denominator 100.
  3. Every integer greater than 1 factors into primes in exactly one way, up to order.
  4. The Euclidean algorithm finds a greatest common divisor without listing factors.

Formulas

Sum of fractions

Add by writing both fractions over a common denominator, then add the numerators.

a/b, c/d
Fractions, with b and d not zero

Percent of a whole

Percent means per hundred. A rate of p percent is the fraction p/100.

part
The piece you want
whole
The reference amount

Euclidean algorithm

Replace the larger number by the remainder until the remainder is zero. The last nonzero remainder is the greatest common divisor.

a mod b
Remainder when a is divided by b

Scientific notation

Write a number as a significand between 1 and 10 times a power of ten. Adding the exponents multiplies the numbers.

m
Significand
k
Integer exponent; negative means a small number

Worked examples

1. Add unlike fractions

Compute 3/4 + 5/6.

  1. Cross-multiply for the numerators: 3×6 = 18 and 5×4 = 20.
  2. Common denominator: 4×6 = 24.
  3. Sum: (18 + 20)/24 = 38/24.
  4. Cancel a factor of 2: 19/12.

19/12, which is 1 and 7/12.

2. A discount is a multiplication

A coat costs $80 and is reduced by 15%. What do you pay?

  1. 15% of 80 is 0.15 × 80 = 12.
  2. Paying 15% less means paying 85% of the price: 0.85 × 80 = 68.
  3. Either route is the same: 80 − 12 = 68.

You pay 68.

Try these

Use Euclid to find gcd(252, 198).

Write 0.00047 in scientific notation.

Common mistakes

  • Adding numerators and denominators separately: 1/2 + 1/3 is not 2/5.
  • A percent of a percent multiplies. A 10% reduction followed by a 10% reduction is not a 20% reduction; it is 0.9 × 0.9 = 0.81, a 19% reduction.
  • Division by zero is undefined. A denominator is never a placeholder you can ignore.