Foundations

Study Sheet

Foundations

Everything you need before Algebra 1

Types of Real Numbers

Concept
The Number Families

Every number you will use in Algebra 1 is a real number. Real numbers sort into nested families:

  • Natural (counting) numbers: 1,2,3,4,…1, 2, 3, 4, \dots
  • Whole numbers: the naturals plus zero: 0,1,2,3,…0, 1, 2, 3, \dots
  • Integers: whole numbers and their negatives: …,−3,−2,−1,0,1,2,3,…\dots, -3, -2, -1, 0, 1, 2, 3, \dots
  • Rational numbers: any number that can be written as a fraction ab\frac{a}{b} where aa and bb are integers and b≠0b \ne 0. Their decimals either end or repeat.
  • Irrational numbers: real numbers that cannot be written as a fraction. Their decimals go on forever with no repeating pattern.

Each family sits inside the next: every natural number is whole, every whole number is an integer, every integer is rational.

Algebra 1 begins by making the arithmetic laws explicit — commutative, associative, distributive — because every simplification is one of them in disguise. Real numbers split into rational and irrational; the order of operations and absolute value carry over from Pre-Algebra. The goal is fluency: rewriting expressions without changing their value.

Each family was added to make one more operation always possible. Whole numbers are closed under addition but not subtraction (3−53 - 5 leaves them), so integers were added; integers are not closed under division (1÷21 \div 2 leaves them), so rationals were added; and lengths like the diagonal of a unit square, 2\sqrt2, are not rational at all, so the reals include the irrationals too. That history is why the families nest.

Example
Sorting numbers into families
real numbers ℝrational ℚ3/52/3integers ℤ−4whole07irrational√2π
Each family sits inside the next; the irrationals are the reals outside ℚ.

Classify each number in every family it belongs to:   7, −4, 0, 35, 0.6‾, 2, π\;7,\ -4,\ 0,\ \tfrac{3}{5},\ 0.\overline{6},\ \sqrt{2},\ \pi.

  • 77: natural, whole, integer, rational
  • −4-4: integer, rational
  • 00: whole, integer, rational
  • 35\tfrac{3}{5}: rational (it is a fraction)
  • 0.6‾=0.6666⋯=230.\overline{6} = 0.6666\dots = \tfrac{2}{3}: rational (repeating decimal)
  • 2=1.41421356…\sqrt{2} = 1.41421356\dots: irrational (never ends or repeats)
  • π=3.14159…\pi = 3.14159\dots: irrational
Side note
Where it goes wrong

Calling every square root irrational. 9=3\sqrt{9} = 3 and 425=25\sqrt{\tfrac{4}{25}} = \tfrac25 are rational; n\sqrt{n} is irrational exactly when nn is not a perfect square (of an integer or a fraction).

Tip

2\sqrt{2} and π\pi are irrational, but 9=3\sqrt{9} = 3 is a perfectly ordinary integer. A square root is only irrational when the number underneath is not a perfect square.

Try it
Try it: every family

List every family for −124-\tfrac{12}{4}, 16\sqrt{16}, 0.16‾0.1\overline{6} and 20\sqrt{20}.

Show the answer

−124=−3-\tfrac{12}{4} = -3: integer, rational, real. 16=4\sqrt{16} = 4: natural, whole, integer, rational, real. 0.16‾=160.1\overline{6} = \tfrac16: rational, real. 20=25\sqrt{20} = 2\sqrt5 (2020 is not a perfect square): irrational, real.

The Number Line & Ordering Numbers

Concept
Reading the Number Line

The number line places every real number in order from small (left) to large (right). For any two numbers, the one farther right is greater.

  • a<ba < b means aa is to the left of bb.
  • a>ba > b means aa is to the right of bb.
  • a≤ba \le b means aa is less than or equal to bb; likewise a≥ba \ge b.

Negatives flip your intuition: the more negative a number looks, the smaller it is. So −8<−3-8 < -3, because −8-8 sits farther left.

Order on the number line is defined so that a<ba < b means b−ab - a is positive, and that is why it behaves the way it does with negatives: −4<−2-4 < -2 because −2−(−4)=2>0-2 - (-4) = 2 > 0. Converting fractions and decimals to one form works because it lets you compare positions directly.

Example
Ordering from least to greatest
−5−4−3−2−101234
Left to right: −4<−2<−12<0<1.5<3-4 < -2 < -\tfrac12 < 0 < 1.5 < 3.

Order   −2, 3, −12, 0, −4, 1.5\;-2,\ 3,\ -\tfrac{1}{2},\ 0,\ -4,\ 1.5.

Place each on the line:−4, −2, −12, 0, 1.5, 3\begin{aligned} \text{Place each on the line:}\quad & -4,\ -2,\ -\tfrac{1}{2},\ 0,\ 1.5,\ 3 \end{aligned}

Least to greatest:  −4<−2<−12<0<1.5<3.\,-4 < -2 < -\tfrac{1}{2} < 0 < 1.5 < 3.

Side note
Where it goes wrong

Ordering −13-\tfrac13 and −0.3-0.3 by the digit: "−0.3<−13-0.3 < -\tfrac13 because 0.3<0.3330.3 < 0.333." Negatives reverse that: −13=−0.333…-\tfrac13 = -0.333\ldots is farther left, so −13<−0.3-\tfrac13 < -0.3.

Tip

When comparing fractions and decimals, rewrite them in the same form. To compare 34\tfrac{3}{4} and 0.70.7, use 34=0.75\tfrac{3}{4} = 0.75, so 34>0.7\tfrac{3}{4} > 0.7.

Try it
Try it: order

Order −73-\tfrac{7}{3}, −2.4-2.4, 5\sqrt{5}, 2.22.2 from least to greatest.

Show the answer

As decimals: −2.333…-2.333\ldots, −2.4-2.4, 2.236…2.236\ldots, 2.22.2. So −2.4<−73<2.2<5-2.4 < -\tfrac73 < 2.2 < \sqrt5.

Absolute Value

Concept
Distance from Zero

The absolute value of a number is its distance from 00 on the number line. Distance is never negative, so absolute value is never negative.

∣5∣=5,∣−5∣=5,∣0∣=0.|5| = 5, \qquad |-5| = 5, \qquad |0| = 0.

A number and its opposite always have the same absolute value.

Absolute value is a distance, and distances ignore direction; that is the whole reason ∣−5∣=∣5∣|-5| = |5|. Treating bars as grouping symbols follows from the same view: you cannot measure how far a quantity is from 00 until you know what the quantity is, so the inside is simplified first.

Example
Absolute value inside expressions

Evaluate   ∣−7∣+∣3∣−∣−2∣\;|-7| + |3| - |{-2}|.

∣−7∣+∣3∣−∣−2∣=7+3−2=8.\begin{aligned} |-7| + |3| - |{-2}| &= 7 + 3 - 2 \\ &= 8. \end{aligned}

Treat each absolute value like grouping: simplify inside first, then use the result.

Side note
Where it goes wrong

Distributing across the bars: ∣−3+8∣=∣−3∣+∣8∣=11|{-3} + 8| = |{-3}| + |8| = 11. Simplify inside first: ∣5∣=5|5| = 5. Absolute value splits over products (∣ab∣=∣a∣∣b∣|ab| = |a||b|) but not over sums.

Tip

Absolute value bars are not the same as parentheses. −∣6∣=−6-|6| = -6, because you take ∣6∣=6|6| = 6 first and then apply the negative sign out front.

Try it
Try it: bars as grouping

Evaluate ∣2−9∣−∣−4∣⋅2|2 - 9| - |{-4}| \cdot 2 and −∣3−32∣-|3 - 3^2|.

Show the answer

∣2−9∣=∣−7∣=7|2 - 9| = |{-7}| = 7 and ∣−4∣⋅2=8|{-4}| \cdot 2 = 8, so 7−8=−17 - 8 = -1. For the second, 3−9=−63 - 9 = -6, ∣−6∣=6|{-6}| = 6, and the outside sign gives −6-6.

Operations with Integers (Sign Rules)

Concept
Adding & Subtracting Signed Numbers

Adding:

  • Same signs: add the values, keep the sign. (−4)+(−6)=−10(-4) + (-6) = -10.
  • Different signs: subtract the smaller value from the larger, keep the sign of the larger.   (−9)+4=−5\;(-9) + 4 = -5.

Subtracting: “add the opposite.” Change subtraction to addition and flip the sign of the second number:

a−b=a+(−b),5−(−3)=5+3=8.a - b = a + (-b), \qquad 5 - (-3) = 5 + 3 = 8.

The sign rules for multiplying are forced by the distributive property: since (−6)(3+(−3))=(−6)(0)=0(-6)(3 + (-3)) = (-6)(0) = 0, we need (−6)(3)+(−6)(−3)=0(-6)(3) + (-6)(-3) = 0, and (−6)(3)=−18(-6)(3) = -18 (six groups of −3-3), so (−6)(−3)(-6)(-3) must be 1818. Division inherits the same rules, because it undoes multiplication.

Concept
Multiplying & Dividing Signed Numbers

The rule is the same for both operations:

  • Same signs →\rightarrow positive answer.
  • Different signs →\rightarrow negative answer.
(−6)×(−3)=18,(−6)×3=−18,−204=−5,−20−5=4.(-6)\times(-3) = 18, \qquad (-6)\times 3 = -18, \qquad \frac{-20}{4} = -5, \qquad \frac{-20}{-5} = 4.
Example
Mixed integer practice

Evaluate   −8+5−(−3)\;-8 + 5 - (-3) and   (−4)(−2)(−1)\;(-4)(-2)(-1).

−8+5−(−3)=−8+5+3=0.(−4)(−2)(−1)=(8)(−1)=−8.\begin{aligned} -8 + 5 - (-3) &= -8 + 5 + 3 = 0. \\[4pt] (-4)(-2)(-1) &= (8)(-1) = -8. \end{aligned}

An odd number of negative factors gives a negative product; an even number gives a positive product.

Side note
Where it goes wrong

Using "two negatives make a positive" for addition: −8+(−3)=11-8 + (-3) = 11. That rule is for multiplication. Adding two negatives moves further left: −8+(−3)=−11-8 + (-3) = -11.

Tip

The trickiest spot is subtracting a negative:   7−(−2)\;7 - (-2) becomes 7+2=97 + 2 = 9. “Minus a minus is a plus.”

Try it
Try it: signs

Evaluate −12−(−5)+(−7)-12 - (-5) + (-7) and (−3)(−8)−6\dfrac{(-3)(-8)}{-6}.

Show the answer

−12+5−7=−14-12 + 5 - 7 = -14. For the second: (−3)(−8)=24(-3)(-8) = 24 (even number of negatives), and 24−6=−4\tfrac{24}{-6} = -4.

Fractions

Concept
Simplifying Fractions

A fraction is in lowest terms when the numerator and denominator share no common factor except 11. Divide both by their greatest common factor (GCF).

1218=12÷618÷6=23.\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}.

Every fraction rule traces back to one fact: ab\tfrac{a}{b} means a÷ba \div b. Multiplying top and bottom by the same number multiplies by 11, which is why equivalent fractions and common denominators work; dividing by cd\tfrac{c}{d} is multiplying by dc\tfrac{d}{c} because cd⋅dc=1\tfrac cd \cdot \tfrac dc = 1.

Concept
The Four Operations
  • Multiply: multiply straight across, then simplify.   23×45=815\;\dfrac{2}{3}\times\dfrac{4}{5} = \dfrac{8}{15}.
  • Divide: multiply by the reciprocal (flip the second fraction).   23÷45=23×54=1012=56\;\dfrac{2}{3}\div\dfrac{4}{5} = \dfrac{2}{3}\times\dfrac{5}{4} = \dfrac{10}{12} = \dfrac{5}{6}.
  • Add / Subtract: rewrite with a common denominator first, then add or subtract the numerators.
14+23=312+812=1112.\frac{1}{4} + \frac{2}{3} = \frac{3}{12} + \frac{8}{12} = \frac{11}{12}.
Concept
Fractions ↔\leftrightarrow Decimals
  • Fraction to decimal: divide the numerator by the denominator.   38=3÷8=0.375\;\tfrac{3}{8} = 3 \div 8 = 0.375.
  • Decimal to fraction: write the decimal over its place value, then simplify.   0.35=35100=720\;0.35 = \tfrac{35}{100} = \tfrac{7}{20}.
Example
A full fraction subtraction

Compute   56−34\;\dfrac{5}{6} - \dfrac{3}{4}.

56−34=1012−912common denominator 12=112.\begin{aligned} \frac{5}{6} - \frac{3}{4} &= \frac{10}{12} - \frac{9}{12} && \text{common denominator } 12 \\ &= \frac{1}{12}. \end{aligned}
Side note
Where it goes wrong

Cancelling terms instead of factors: 2+62=6\tfrac{2 + 6}{2} = 6. Only a factor of the whole numerator cancels: 2+62=82=4\tfrac{2 + 6}{2} = \tfrac{8}{2} = 4, or 2(1+3)2=4\tfrac{2(1 + 3)}{2} = 4.

Tip

Only add or subtract fractions after finding a common denominator, but for multiplying and dividing you do not need one. Mixing these up is the most common fraction mistake.

Try it
Try it: all four operations

Compute 23+58\dfrac{2}{3} + \dfrac{5}{8}, 23−58\dfrac{2}{3} - \dfrac{5}{8}, 23⋅58\dfrac{2}{3} \cdot \dfrac{5}{8} and 23÷58\dfrac{2}{3} \div \dfrac{5}{8}.

Show the answer

LCD 2424: 1624+1524=3124\tfrac{16}{24} + \tfrac{15}{24} = \tfrac{31}{24} and 1624−1524=124\tfrac{16}{24} - \tfrac{15}{24} = \tfrac{1}{24}. Product: 1024=512\tfrac{10}{24} = \tfrac{5}{12}. Quotient: 23⋅85=1615\tfrac23 \cdot \tfrac85 = \tfrac{16}{15}.

Order of Operations (PEMDAS)

Concept
The Order

Simplify in this order:

  • P — Parentheses / grouping (including absolute value and fraction bars)
  • E — Exponents
  • MD — Multiplication and Division, left to right
  • AS — Addition and Subtraction, left to right

Multiplication and division rank equally; do whichever comes first reading left to right. Same for addition and subtraction.

The order is a shared convention that gives every expression one value. It follows the strength of the operations: exponents are repeated multiplication and bind tightest, multiplication is repeated addition and binds next. Multiplication and division tie because dividing is multiplying by a reciprocal, and addition and subtraction tie because subtracting is adding an opposite, so within a tie you simply read left to right.

Example
Working through PEMDAS

Evaluate   3+2×(5−1)2÷8\;3 + 2 \times (5 - 1)^2 \div 8.

3+2×(5−1)2÷8=3+2×(4)2÷8parentheses=3+2×16÷8exponent=3+32÷8multiply (left first)=3+4divide=7.add\begin{aligned} 3 + 2 \times (5-1)^2 \div 8 &= 3 + 2 \times (4)^2 \div 8 && \text{parentheses} \\ &= 3 + 2 \times 16 \div 8 && \text{exponent} \\ &= 3 + 32 \div 8 && \text{multiply (left first)} \\ &= 3 + 4 && \text{divide} \\ &= 7. && \text{add} \end{aligned}
Side note
Where it goes wrong

Doing multiplication before division when division comes first: 24÷4×224 \div 4 \times 2 as 24÷8=324 \div 8 = 3. They tie, so left to right: 6×2=126 \times 2 = 12.

Tip

2×16÷82 \times 16 \div 8 is done left to right: 32÷8=432 \div 8 = 4. Do not do all multiplication before all division — they share a step.

Try it
Try it: evaluate

Evaluate 18−32⋅2+12÷(1−4)18 - 3^2 \cdot 2 + 12 \div (1 - 4).

Show the answer

Parentheses: 1−4=−31 - 4 = -3. Exponent: 32=93^2 = 9. Multiply and divide left to right: 9⋅2=189 \cdot 2 = 18 and 12÷(−3)=−412 \div (-3) = -4. Then 18−18+(−4)=−418 - 18 + (-4) = -4.

Properties of Operations

Concept
The Named Properties
  • Commutative (order): a+b=b+aa + b = b + a and a⋅b=b⋅aa \cdot b = b \cdot a.
  • Associative (grouping): (a+b)+c=a+(b+c)(a+b)+c = a+(b+c) and (ab)c=a(bc)(ab)c = a(bc).
  • Distributive (spread over a sum): a(b+c)=ab+aca(b+c) = ab + ac.
  • Identity: adding 00 or multiplying by 11 changes nothing: a+0=aa + 0 = a, a⋅1=aa \cdot 1 = a.

These properties are the complete list of rewrites that can never change a value, so every simplification in algebra is a chain of them. The distributive property is the one that links multiplication and addition, and it is a fact about area: a 44-by-(x+3)(x + 3) rectangle is a 44-by-xx piece next to a 44-by-33 piece.

Example
Using the distributive property
4x12x344(x + 3) = 4x + 12
One rectangle measured two ways.

Expand   4(x+3)\;4(x + 3) and simplify   −2(3y−5)\;-2(3y - 5).

4(x+3)=4⋅x+4⋅3=4x+12.−2(3y−5)=(−2)(3y)+(−2)(−5)=−6y+10.\begin{aligned} 4(x+3) &= 4\cdot x + 4\cdot 3 = 4x + 12. \\[4pt] -2(3y-5) &= (-2)(3y) + (-2)(-5) = -6y + 10. \end{aligned}
Side note
Where it goes wrong

Distributing over multiplication: 2(3x)=6⋅2x2(3x) = 6 \cdot 2x or 2(3⋅x)=6⋅2x2(3 \cdot x) = 6 \cdot 2x. The distributive property spreads over a sum; for a product the associative property applies: 2(3x)=(2⋅3)x=6x2(3x) = (2 \cdot 3)x = 6x.

Tip

Subtraction and division are not commutative: 7−3≠3−77 - 3 \ne 3 - 7. When you distribute a negative, every term inside changes sign.

Try it
Try it: name each step

Justify each step: 3(x+5)+2x=(3x+15)+2x=3x+(15+2x)=3x+(2x+15)=(3x+2x)+15=5x+153(x + 5) + 2x = (3x + 15) + 2x = 3x + (15 + 2x) = 3x + (2x + 15) = (3x + 2x) + 15 = 5x + 15.

Show the answer

Distributive; associative (regroup the additions); commutative (swap 1515 and 2x2x); associative again; then distributive in reverse to combine 3x+2x=(3+2)x=5x3x + 2x = (3 + 2)x = 5x.

Evaluating Expressions by Substitution

Concept
Substitute, then Simplify

To evaluate an algebraic expression, replace each variable with its given value (use parentheses!) and then follow PEMDAS.

A variable is a placeholder for a number, so evaluating is replacing it and computing. Parentheses are what keep the substituted number intact: 3x23x^2 at x=−2x = -2 is 3(−2)2=123(-2)^2 = 12, while writing 3⋅−223 \cdot -2^2 would square only the 22 and give −12-12.

Example
Plugging in values

Evaluate   3x2−2y\;3x^2 - 2y when x=−2x = -2 and y=5y = 5.

3x2−2y=3(−2)2−2(5)=3(4)−10=12−10=2.\begin{aligned} 3x^2 - 2y &= 3(-2)^2 - 2(5) \\ &= 3(4) - 10 \\ &= 12 - 10 = 2. \end{aligned}
Side note
Where it goes wrong

Substituting without parentheses: 3x2−2y3x^2 - 2y at x=−2x = -2 written as 3⋅−22−2⋅5=−12−10=−223 \cdot -2^2 - 2 \cdot 5 = -12 - 10 = -22. With parentheses, 3(−2)2−2(5)=12−10=23(-2)^2 - 2(5) = 12 - 10 = 2.

Tip

Wrap every substituted value in parentheses. Without them, (−2)2=4(-2)^2 = 4 can be mistaken for −22=−4-2^2 = -4.

Try it
Try it: evaluate

Evaluate a2−4b2a\dfrac{a^2 - 4b}{2a} at a=−3a = -3, b=−1b = -1.

Show the answer

Substitute with parentheses: (−3)2−4(−1)2(−3)=9+4−6=−136\tfrac{(-3)^2 - 4(-1)}{2(-3)} = \tfrac{9 + 4}{-6} = -\tfrac{13}{6}.

Combining Like Terms

Concept
What Makes Terms “Like”

Like terms have exactly the same variable part (same letters with the same exponents). You may add or subtract like terms by combining their coefficients; the variable part stays the same.

5x+3x=8x,7a−2a=5a.5x + 3x = 8x, \qquad 7a - 2a = 5a.

Constants (plain numbers) are like terms with each other.

Combining like terms is the distributive property backward: 6x−2x−x=(6−2−1)x=3x6x - 2x - x = (6 - 2 - 1)x = 3x. That only works when the terms share the exact same variable part, because that shared part is the common factor being pulled out. x2x^2 and xx share only one xx, so x2+x=x(x+1)x^2 + x = x(x + 1) is as far as they go.

Example
Simplifying an expression

Simplify   6x+4−2x+9−x\;6x + 4 - 2x + 9 - x.

6x+4−2x+9−x=(6x−2x−x)+(4+9)=3x+13.\begin{aligned} 6x + 4 - 2x + 9 - x &= (6x - 2x - x) + (4 + 9) \\ &= 3x + 13. \end{aligned}
Side note
Where it goes wrong

Combining unlike powers: 3x2+2x=5x33x^2 + 2x = 5x^3. Test with x=1x = 1: the left side is 55 and the right side is 55, but with x=2x = 2 the left side is 1616 and the right side is 4040. They are different terms and stay separate.

Tip

x2x^2 and xx are not like terms, so x2+xx^2 + x cannot be combined. Keep the exponents in mind before merging.

Try it
Try it: simplify

Simplify 4a2−3a+7−a2+5a−104a^2 - 3a + 7 - a^2 + 5a - 10.

Show the answer

Group by variable part: a2a^2-terms 4a2−a2=3a24a^2 - a^2 = 3a^2; aa-terms −3a+5a=2a-3a + 5a = 2a; constants 7−10=−37 - 10 = -3. Result: 3a2+2a−33a^2 + 2a - 3.

Going Deeper: Advanced Foundations

Concept
Closure & the Field Properties of the Reals

A set of numbers is closed under an operation if combining any two members always lands you back inside that set.

  • The integers are closed under ++, −-, and ×\times, but not under ÷\div:   1÷2=12\;1 \div 2 = \tfrac{1}{2} escapes the integers.
  • The real numbers are closed under ++, −-, ×\times, and ÷\div (dividing by 00 excepted).

Because the reals also obey commutativity, associativity, and distributivity, carry additive and multiplicative identities (00 and 11), and give every number an additive inverse −a-a and every nonzero number a multiplicative inverse 1a\tfrac{1}{a}, mathematicians call R\mathbb{R} a field. These field properties are the rules that justify every algebra step you take.

Tip

Closure is about staying inside a set, not about getting a “nice” answer. The naturals are not closed under subtraction because 3−5=−23 - 5 = -2 is not natural, even though −2-2 is a fine number.

Concept
Absolute Value as Distance: ∣a−b∣|a - b|
−4−3−2−1012345|−3 − 4| = 7
The distance between −3-3 and 44 is 77, in either order.

Beyond “distance from 00,” absolute value measures the distance between any two numbers:

∣a−b∣=the distance between a and b on the number line.|a - b| = \text{the distance between } a \text{ and } b \text{ on the number line}.

Order does not matter, because ∣a−b∣=∣b−a∣|a - b| = |b - a|. For example, the gap between −3-3 and 44 is

∣−3−4∣=∣−7∣=7and equally∣4−(−3)∣=∣7∣=7.|-3 - 4| = |{-7}| = 7 \qquad\text{and equally}\qquad |4 - (-3)| = |7| = 7.

This is why ∣x−5∣=2|x - 5| = 2 describes every point exactly 22 units from 55, namely x=3x = 3 and x=7x = 7.

Example
Nested absolute value

Evaluate   ∣ 3−∣−8∣ ∣+∣ 2⋅∣−4∣−11 ∣\;\bigl|\,3 - |{-8}| \,\bigr| + \bigl|\,2 \cdot |{-4}| - 11 \,\bigr|.

∣ 3−∣−8∣ ∣+∣ 2⋅∣−4∣−11 ∣=∣ 3−8 ∣+∣ 2⋅4−11 ∣innermost bars first=∣−5∣+∣ 8−11 ∣simplify inside=∣−5∣+∣−3∣=5+3=8.\begin{aligned} \bigl|\,3 - |{-8}| \,\bigr| + \bigl|\,2\cdot|{-4}| - 11 \,\bigr| &= \bigl|\,3 - 8 \,\bigr| + \bigl|\,2\cdot 4 - 11 \,\bigr| && \text{innermost bars first} \\ &= |{-5}| + |\,8 - 11\,| && \text{simplify inside} \\ &= |{-5}| + |{-3}| \\ &= 5 + 3 = 8. \end{aligned}

Work an absolute value from the inside out, exactly like nested parentheses.

Concept
The Real-Number Hierarchy N⊂Z⊂Q⊂R\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}

The number families nest as a chain of subsets, each written with its own symbol:

N⊂Z⊂Q⊂R,\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R},

that is, naturals ⊂\subset integers ⊂\subset rationals ⊂\subset reals. The irrationals are the reals that are left over when you remove the rationals — they fill every gap the fractions miss, which is why the number line has no holes.

Example
Why 2\sqrt{2} is irrational (a proof sketch)

Suppose, for contradiction, that 2\sqrt{2} were rational. Then we could write it in lowest terms:

2=ab,a,b integers with no common factor, b≠0.\sqrt{2} = \frac{a}{b}, \qquad a,b \text{ integers with no common factor}, \ b \ne 0.

Squaring both sides and clearing the denominator:

2=a2b2a2=2b2.\begin{aligned} 2 &= \frac{a^2}{b^2} \\ a^2 &= 2b^2. \end{aligned}

So a2a^2 is even, which forces aa itself to be even; write a=2ka = 2k. Substituting back:

(2k)2=2b2  ⟹  4k2=2b2  ⟹  b2=2k2,(2k)^2 = 2b^2 \;\Longrightarrow\; 4k^2 = 2b^2 \;\Longrightarrow\; b^2 = 2k^2,

so bb is even too. But then aa and bb share the factor 22, contradicting “lowest terms.” The assumption must be false, so 2\sqrt{2} cannot be written as a fraction — it is irrational.

Concept
Efficient Order of Operations with Nested Grouping

When grouping symbols are nested — parentheses inside brackets inside braces, or a fraction bar acting as grouping above and below — always resolve the innermost group first and work outward. A fraction bar groups its entire numerator and entire denominator as if each were parenthesized.

2+3⋅47−3 means (2+3⋅4)(7−3)=144=72.\frac{2+3\cdot 4}{7-3} \ \text{means}\ \frac{(2+3\cdot 4)}{(7-3)} = \frac{14}{4} = \frac{7}{2}.
Example
Peeling nested brackets

Evaluate   20−[ 3+2(6−4)2 ]\;20 - \bigl[\,3 + 2\left(6 - 4\right)^2\,\bigr].

20−[ 3+2(6−4)2 ]=20−[ 3+2(2)2 ]innermost parentheses=20−[ 3+2⋅4 ]exponent inside the bracket=20−[ 3+8 ]multiply=20−11=9.\begin{aligned} 20 - \bigl[\,3 + 2(6-4)^2\,\bigr] &= 20 - \bigl[\,3 + 2(2)^2\,\bigr] && \text{innermost parentheses} \\ &= 20 - \bigl[\,3 + 2\cdot 4\,\bigr] && \text{exponent inside the bracket} \\ &= 20 - \bigl[\,3 + 8\,\bigr] && \text{multiply} \\ &= 20 - 11 = 9. \end{aligned}
Concept
A Preview of Summation (Σ\Sigma) Notation

The Greek capital sigma Σ\Sigma is shorthand for “add up a list.” The expression

∑i=1nai  =  a1+a2+a3+⋯+an\sum_{i=1}^{n} a_i \;=\; a_1 + a_2 + a_3 + \dots + a_n

reads: “start the counter ii at 11, plug it into the rule after the Σ\Sigma, and add the results up to i=ni = n.” The letter ii is just an index; it disappears once the sum is computed. For instance,

∑k=14(2k−1)=(2⋅1−1)+(2⋅2−1)+(2⋅3−1)+(2⋅4−1)=1+3+5+7=16.\sum_{k=1}^{4} (2k-1) = (2\cdot 1 - 1) + (2\cdot 2 - 1) + (2\cdot 3 - 1) + (2\cdot 4 - 1) = 1 + 3 + 5 + 7 = 16.

You will meet Σ\Sigma again with sequences and series — for now, just read it as a compact “add these terms.”

Example
Evaluating a tricky multi-variable expression

Evaluate   a2−b c∣ b−a ∣\;\dfrac{a^2 - b\,c}{|\,b - a\,|} when a=−3a = -3, b=4b = 4, and c=−2c = -2.

a2−bc∣ b−a ∣=(−3)2−(4)(−2)∣ 4−(−3) ∣substitute, each value in parentheses=9−(−8)∣ 7 ∣exponent and product on top=9+87subtracting a negative=177.\begin{aligned} \frac{a^2 - bc}{|\,b - a\,|} &= \frac{(-3)^2 - (4)(-2)}{|\,4 - (-3)\,|} && \text{substitute, each value in parentheses} \\ &= \frac{9 - (-8)}{|\,7\,|} && \text{exponent and product on top} \\ &= \frac{9 + 8}{7} && \text{subtracting a negative} \\ &= \frac{17}{7}. \end{aligned}

Simplify the numerator and denominator separately (the fraction bar groups each), then divide.

Tip

With several variables, substitute every occurrence at once and wrap each value in parentheses before simplifying. A single missed sign — like reading −b c-b\,c as positive — derails the whole evaluation.

Why It Works

Two expressions are equivalent when they agree for every value of the variable, and the laws of arithmetic are exactly the guarantees that a rewrite keeps that agreement. That gives a practical proof technique: to check whether (x+3)2(x + 3)^2 equals x2+9x^2 + 9, substitute x=1x = 1 — 16≠1016 \ne 10, so they are not equivalent. A single counterexample settles it; agreement at one value does not prove equivalence, but disagreement disproves it.

Tip
Reading a proof

Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.

Formulas, Proofs & Tips

Tip
Order of operations
Parentheses→Exponents→Multiply/Divide→Add/Subtract\text{Parentheses} \to \text{Exponents} \to \text{Multiply/Divide} \to \text{Add/Subtract}

What it means. A fixed order so every reader of an expression gets the same value.

Example. 3+4×2=3+8=113+4\times2=3+8=11 (multiply before adding), not 1414.

Why it works. Multiplication is repeated addition and exponents are repeated multiplication, so each level is a shorthand for the one below it and must be unpacked first. Parentheses override the order by grouping explicitly.

Tip. Multiply/divide are one level worked left to right, and so are add/subtract. 8−3+28-3+2 is 77, not 33.

Tip
The distributive property
a(b+c)=ab+aca(b+c)=ab+ac

What it means. Multiplying a sum multiplies each piece of it.

Example. 3(x+2)=3x+63(x+2)=3x+6.

Why it works. a(b+c)a(b+c) is b+cb+c added aa times. Regrouping those copies gives bb added aa times plus cc added aa times, i.e. ab+acab+ac. It is also the area of an a×(b+c)a\times(b+c) rectangle split into two.

Tip. Distribute the sign too: −(x−4)=−x+4-(x-4)=-x+4. Run it backwards to factor.

Tip
Solving equations and inequalities
Do the same thing to both sides;multiplying an inequality by a negative flips it\text{Do the same thing to both sides};\qquad \text{multiplying an inequality by a negative flips it}

What it means. Equations stay balanced under identical operations; inequalities have one extra rule.

Example. 2x+3=11⇒2x=8⇒x=42x+3=11\Rightarrow 2x=8\Rightarrow x=4.

Why it works. If a=ba=b then a+c=b+ca+c=b+c and ac=bcac=bc — equality is preserved. But if a<ba<b, multiplying by −1-1 reverses their order on the number line, since −a-a now lies to the right of −b-b.

Tip. Undo operations in reverse PEMDAS order: addition first, multiplication last.