Numbers & Operations
21 concepts
Place Value
A digit's value depends on which spot (place) it stands in.
What a Fraction Is
A number that tells how many equal parts the whole is cut into (bottom) and how many you have (top).
What a Decimal Is
A number that adds places to the right of a dot (the decimal point) to write parts smaller than one.
a/c ± b/c = (a ± b)/cAdding & Subtracting Fractions
When the slices are the same size (same bottom), just add or subtract how many slices (the tops).
Multiplication & Division
Multiplication is adding the same number many times; division is sharing equally or counting how many groups.
a : b = a ÷ bRatio & Rate
A ratio compares two amounts as 'this to that'; a rate turns that into one number by dividing.
p = (a ÷ b) × 100Percent
A number that says how much the part is when the whole is treated as 100. The sign % means 'out of 100'.
N = p₁^a₁ × p₂^a₂ × … × pₖ^aₖPrime Factorization
A single number broken down into a product of primes only — 2, 3, 5, 7, and so on.
Integers on the Number Line
The positive integers, 0, and the negative integers laid out in order on one line. Bigger to the right, smaller to the left.
|a| = a (a ≥ 0), |a| = −a (a < 0)Absolute Value
How far a number is from 0. Throw away the direction (the sign) and keep only the distance.
(+)×(+)=(+), (−)×(−)=(+), (+)×(−)=(−)Arithmetic with Rational Numbers
The rules for adding, subtracting, multiplying, and dividing integers and fractions. The key is to settle the sign first, then compute the sizes.
aᵐ · aⁿ = aᵐ⁺ⁿLaws of Exponents
A power is the same number multiplied several times. Multiply or divide those, and you just add or subtract the exponents.
√a = b ⇔ b² = a (a ≥ 0, b ≥ 0)Square Roots
When some number squared gives a, that number is a square root of a. √a is the one that is 0 or positive.
Irrational & Real Numbers
Numbers you can write as a fraction (integer/integer) are rational; those you can't are irrational. Together they form the reals.
0.ddd… = d/9, 0.abab… = ab/99Repeating Decimals
A decimal where the same block of digits repeats forever after the point. It can always be turned back into a fraction.
GCD(a, b) × LCM(a, b) = a × bGCD and LCM
The GCD is the largest number dividing both; the LCM is the smallest number that is a multiple of both — the first place their multiples overlap.
i² = −1, z = a + biComplex Numbers
Introduce a new number i whose square is −1, and every number can be written as a+bi, a real part a plus an imaginary part bi. That's a complex number.
(G, ∗): ① a∗b ∈ G ② (a∗b)∗c = a∗(b∗c) ③ a∗e = e∗a = a ④ a∗a⁻¹ = eGroups: The Language of Symmetry
A group is a set plus an operation, where the operation obeys four rules: (1) results never leak out (closure), (2) grouping doesn't matter (associativity), (3) there is a do-nothing identity, and (4) every element has an inverse that undoes it.
a ≡ b (mod n) ⟺ n | (a − b)Modular Arithmetic
Modular arithmetic is arithmetic that wraps back to 0 after some number n. Two numbers are 'congruent' if they leave the same remainder when divided by n, written a ≡ b (mod n). Only the remainder matters.
e^(iθ) = cos θ + i sin θ, e^(iπ) + 1 = 0The Complex Plane & Euler's Formula
See a complex number a+bi as a point (or arrow) in the plane, and you get the complex plane. Euler's formula e^(iθ)=cos θ+i sin θ says that the complex exponential e^(iθ) is the point on the unit circle rotated by angle θ. In short: an imaginary exponent means rotation.
|ℕ| = |ℤ| = |ℚ| = ℵ₀ < |ℝ|Countable Infinity
If the elements of an infinite set can be numbered 1,2,3,… with none left out, it is 'countably infinite'. The naturals, integers, and rationals all qualify. Yet the reals cannot be numbered that way even in principle — they are a larger infinity.
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