Integers

Study Sheet

Integers

Everything you need to know about negative numbers

What Integers Are

Concept
The Big Idea
−8−6−4−2024687 steps7 steps
−7-7 and 77 are opposites: the same distance from 00 on opposite sides.

The integers are the whole numbers together with their negatives and zero:

…, −3, −2, −1, 0, 1, 2, 3, …\ldots,\ -3,\ -2,\ -1,\ 0,\ 1,\ 2,\ 3,\ \ldots

Integers do not include fractions or decimals. A positive integer is greater than 00, a negative integer is less than 00, and 00 itself is neither positive nor negative.

Integers are the whole numbers and their negatives. Adding integers is combining moves on the number line; subtracting is adding the opposite; multiplying and dividing follow the sign rule (same signs positive, different signs negative). Absolute value gives the size of an integer without its sign.

Negative numbers exist so that subtraction always has an answer. With only 0,1,2,…0, 1, 2, \ldots, the problem 3−53 - 5 has nowhere to land; extending the number line past 00 gives it one, −2-2. Every negative number is paired with a positive one at the same distance from 00, its opposite, and a number plus its opposite is always 00.

The Number Line

Every integer has a home on the number line. Positive numbers sit to the right of 00, negative numbers sit to the left. As you move right, numbers get bigger; as you move left, numbers get smaller.

Opposites

Two numbers are opposites if they are the same distance from 00 but on opposite sides. The opposite of 55 is −5-5, and the opposite of −8-8 is 88. The opposite of 00 is 00.

Example
Finding Opposites

The opposite of −7-7 is 77, because both are 77 steps from 00 (one left, one right). Writing a negative sign in front means “the opposite of”: −(−3)=3-(-3) = 3. Two opposites in a row cancel out.

Side note
Where it goes wrong

Thinking −(−3)-(-3) is negative because "it has minus signs." Each minus sign means "the opposite of": the opposite of −3-3 is 33. Two opposites in a row bring you back where you started.

Tip

Tip: A negative sign means “opposite of.” So −(−4)-(-4) reads as “the opposite of −4-4,” which is 44.

Try it
Try it: opposites

Write the opposite of each: 1212, −5-5, 00, and −(−8)-(-8).

Show the answer

The opposite of 1212 is −12-12; of −5-5 is 55; of 00 is 00 (zero is its own opposite). For the last one, first simplify: −(−8)=8-(-8) = 8, and the opposite of 88 is −8-8.

Absolute Value

Concept
Definition
−7−6−5−4−3−2−101234567|−6| = 6|6| = 6
Six steps from 00 either way: ∣−6∣=∣6∣=6|-6| = |6| = 6.

The absolute value of a number is its distance from 00 on the number line. Distance is never negative, so absolute value is always 00 or positive. We write it with bars: ∣−6∣=6|{-6}| = 6 and ∣6∣=6|6| = 6.

Absolute value strips away direction and keeps size. On the number line, −6-6 and 66 are the same distance from 00, six steps, and distance does not care which way you walked. That is why ∣a∣|a| is never negative, and why ∣a∣=∣−a∣|a| = |{-a}| for every number aa.

Example
Working with Absolute Value

∣−9∣=9|{-9}| = 9   ∣12∣=12|12| = 12   ∣0∣=0|0| = 0 Absolute value tells you how far, not which direction. Both −9-9 and 99 are 99 units from zero, so both have absolute value 99.

Side note
Where it goes wrong

Treating absolute value as "make it positive" even outside the bars: −∣−5∣=5-|{-5}| = 5. The bars act first, giving ∣−5∣=5|{-5}| = 5; the minus sign outside then makes it −5-5.

Tip

Tip: To find ∣x∣|x|, just drop the sign. But be careful: a negative sign outside the bars stays, so −∣−5∣=−(5)=−5-|{-5}| = -(5) = -5.

Try it
Try it: evaluate

Evaluate ∣−14∣|{-14}|, ∣3−10∣|3 - 10|, and −∣−2∣+∣−9∣-|{-2}| + |{-9}|.

Show the answer

∣−14∣=14|{-14}| = 14. For ∣3−10∣|3 - 10|, work inside first: 3−10=−73 - 10 = -7, and ∣−7∣=7|{-7}| = 7. For the last: ∣−2∣=2|{-2}| = 2 and ∣−9∣=9|{-9}| = 9, so −2+9=7-2 + 9 = 7.

Comparing and Ordering Integers

Concept
The Rule

On the number line, the number farther to the right is greater. So any positive number is greater than any negative number, and for two negatives, the one closer to 00 is greater. That means −2>−7-2 > -7, because −2-2 is to the right of −7-7.

On a number line, "greater" means "farther right," and that single rule settles every comparison. It explains the surprising cases: −2>−7-2 > -7 because −2-2 is to the right, even though 77 is bigger than 22. Among negatives, the one with the larger absolute value is farther from zero on the left, and so it is the smaller number.

Example
Ordering from Least to Greatest
−6−5−4−3−2−1012345
Read left to right: least to greatest.

Order 3, −5, 0, −1, 43,\ -5,\ 0,\ -1,\ 4 from least to greatest. Picture them on the number line and read left to right:

−5<−1<0<3<4.-5 < -1 < 0 < 3 < 4.
Side note
Where it goes wrong

Ordering negatives by their digits: "−3<−8-3 < -8 because 3<83 < 8." On the number line −8-8 is farther left, so −8<−3-8 < -3.

Example
Real-World Context: Temperature

On a cold day the temperatures were −3∘-3^\circ, 5∘5^\circ, −8∘-8^\circ, and 0∘0^\circ. Which is coldest? Coldest means smallest (farthest left): −8∘-8^\circ. Warmest is 5∘5^\circ. In order from coldest to warmest: −8∘, −3∘, 0∘, 5∘-8^\circ,\ -3^\circ,\ 0^\circ,\ 5^\circ.

Tip

Tip: With negatives, “bigger digits” can be smaller numbers. −100-100 is much less than −2-2. Always think about position on the line.

Try it
Try it: order

Order from least to greatest: −12-12, 44, −3-3, 00, −20-20, 77.

Show the answer

Picture them on the line. Farthest left is −20-20, then −12-12, then −3-3, then 00, then 44 and 77: −20<−12<−3<0<4<7-20 < -12 < -3 < 0 < 4 < 7.

Adding Integers

Concept
Two Cases

Same signs: add the absolute values and keep the common sign. Different signs: subtract the smaller absolute value from the larger, and keep the sign of the number with the larger absolute value.

Adding a number means moving along the number line: right for a positive, left for a negative. With the same signs, both moves go the same way, so the distances add. With different signs, the moves go opposite ways and partly cancel; what is left is the difference of the distances, in the direction of the longer move. The two sign rules are just these two pictures.

Example
Same Signs
−11−10−9−8−7−6−5−4−3−2−101−6−4
Start at 00, move 66 left, then 44 more left: −10-10.

−6+(−4)-6 + (-4): both negative, so add 6+4=106 + 4 = 10 and keep the negative sign:  −10\ -10. 7+5=127 + 5 = 12: both positive, answer stays positive.

Side note
Where it goes wrong

Applying the multiplication sign rule to addition: "two negatives make a positive," so −4+(−6)=10-4 + (-6) = 10. In addition, two negatives move you twice to the left: −4+(−6)=−10-4 + (-6) = -10.

Example
Different Signs
−10−9−8−7−6−5−4−3−2−101−9+4
99 left, then 44 back right: −9+4=−5-9 + 4 = -5.

−9+4-9 + 4: subtract 9−4=59 - 4 = 5; since 9>49 > 4, keep the negative sign:  −5\ -5. 8+(−3)8 + (-3): subtract 8−3=58 - 3 = 5; since 8>38 > 3, keep the positive sign:  5\ 5.

Tip

Sign rule for adding: same signs →\rightarrow add and keep the sign; different signs →\rightarrow subtract and take the sign of the “bigger” number.

Try it
Try it: add

Compute −15+8-15 + 8, −7+(−13)-7 + (-13) and 22+(−30)22 + (-30).

Show the answer

−15+8-15 + 8: different signs, 15−8=715 - 8 = 7, and 1515 is the larger distance, so −7-7. −7+(−13)-7 + (-13): same signs, 7+13=207 + 13 = 20, keep the negative: −20-20. 22+(−30)22 + (-30): different signs, 30−22=830 - 22 = 8, and 3030 belongs to the negative: −8-8.

Subtracting Integers

Concept
Add the Opposite

To subtract an integer, add its opposite. Change the subtraction to addition and flip the sign of the number being subtracted:

a−b=a+(−b).a - b = a + (-b).

Then follow the rules for adding integers.

Subtraction asks "what must be added to bb to reach aa?" For 3−(−7)3 - (-7): starting at −7-7, you must move 1010 to the right to reach 33, so the answer is 1010, exactly 3+73 + 7. In general, adding −b-b undoes adding bb, so taking bb away and adding −b-b are the same move. That is the whole reason subtraction becomes "add the opposite."

Example
Subtract Means Add the Opposite
−4−3−2−101234565 − 8 = 5 + (−8)
Subtracting 88 is moving 88 left, the same as adding −8-8.

5−8=5+(−8)=−35 - 8 = 5 + (-8) = -3. −4−6=−4+(−6)=−10-4 - 6 = -4 + (-6) = -10. 3−(−7)=3+(+7)=103 - (-7) = 3 + (+7) = 10.   (Subtracting a negative adds!)

Side note
Where it goes wrong

Changing the wrong sign: −4−6-4 - 6 rewritten as 4+6=104 + 6 = 10. Only the operation and the number AFTER it change: −4−6=−4+(−6)=−10-4 - 6 = -4 + (-6) = -10. The first number keeps its sign.

Tip

Key idea: Subtracting means adding the opposite. A minus sign in front of a negative, like −(−7)-(-7), becomes a plus.

Try it
Try it: subtract

Compute −9−4-9 - 4, 6−(−11)6 - (-11) and −3−(−8)-3 - (-8).

Show the answer

Rewrite each as adding the opposite. −9−4=−9+(−4)=−13-9 - 4 = -9 + (-4) = -13. 6−(−11)=6+11=176 - (-11) = 6 + 11 = 17. −3−(−8)=−3+8=5-3 - (-8) = -3 + 8 = 5.

Multiplying Integers

Concept
Sign Rules for Products

Multiply the absolute values, then decide the sign:

  • positive ×\times positive == positive
  • negative ×\times negative == positive
  • positive ×\times negative == negative (either order)

Same signs give a positive; different signs give a negative.

A positive times a negative is repeated addition: 3×(−7)3 \times (-7) is three jumps of −7-7, landing at −21-21. For a negative times a negative, follow a pattern that has to continue: 3(−2)=−63(-2) = -6, 2(−2)=−42(-2) = -4, 1(−2)=−21(-2) = -2, 0(−2)=00(-2) = 0. Each time the first factor drops by 11, the product rises by 22, so the next line must be (−1)(−2)=2(-1)(-2) = 2. Any other value would break the pattern, and with it the distributive property (the proof is in Going Deeper).

Example
Multiplying
−7−7−7−21−14−70
3×(−7)3 \times (-7): three jumps of −7-7 from 00.

(−6)(−4)=24(-6)(-4) = 24   (same signs →\rightarrow positive) (−7)(3)=−21(-7)(3) = -21   (different signs →\rightarrow negative) (8)(−5)=−40(8)(-5) = -40   (different signs →\rightarrow negative)

Side note
Where it goes wrong

Counting only the first two signs in a longer product: (−2)(−3)(−1)=6(-2)(-3)(-1) = 6. Every negative factor flips the sign. Three negatives give a negative: (−2)(−3)=6(-2)(-3) = 6, then 6×(−1)=−66 \times (-1) = -6.

Tip

Sign rule: an even number of negative factors gives a positive answer; an odd number of negative factors gives a negative answer.

Try it
Try it: signs first

Compute (−4)(9)(-4)(9), (−5)(−6)(-5)(-6) and (−1)(−2)(−3)(−4)(-1)(-2)(-3)(-4).

Show the answer

(−4)(9)=−36(-4)(9) = -36 (different signs). (−5)(−6)=30(-5)(-6) = 30 (same signs). The last has four negative factors, an even number, so it is positive: 1⋅2⋅3⋅4=241 \cdot 2 \cdot 3 \cdot 4 = 24.

Dividing Integers

Concept
Same Sign Rules as Multiplying

Division follows the exact same sign rules as multiplication. Divide the absolute values, then apply: same signs →\rightarrow positive, different signs →\rightarrow negative.

Division is multiplication run backward, so it must follow the same sign rules. −20÷(−4)-20 \div (-4) asks "what times −4-4 gives −20-20?" The answer is 55, because 5×(−4)=−205 \times (-4) = -20; a negative answer would give (−5)(−4)=+20(-5)(-4) = +20, the wrong sign. Every division fact can be checked this way, by multiplying back.

Example
Dividing

−20−4=5\dfrac{-20}{-4} = 5   (same signs →\rightarrow positive) 36−9=−4\dfrac{36}{-9} = -4   (different signs →\rightarrow negative) −306=−5\dfrac{-30}{6} = -5   (different signs →\rightarrow negative)

Side note
Where it goes wrong

Giving 0÷(−5)0 \div (-5) as "undefined" or −5÷0-5 \div 0 as 00. Zero divided by any nonzero number is 00, since 0×(−5)=00 \times (-5) = 0. Dividing BY zero is what is undefined: no number times 00 gives −5-5.

Tip

Tip: Multiplication and division share one motto: same signs →\rightarrow positive, different signs →\rightarrow negative.

Try it
Try it: divide and check

Compute −48÷6-48 \div 6, −56÷(−8)-56 \div (-8) and 90−15\dfrac{90}{-15}, checking each by multiplication.

Show the answer

−48÷6=−8-48 \div 6 = -8; check −8×6=−48-8 \times 6 = -48. −56÷(−8)=7-56 \div (-8) = 7; check 7×(−8)=−567 \times (-8) = -56. 90−15=−6\tfrac{90}{-15} = -6; check −6×(−15)=90-6 \times (-15) = 90.

Order of Operations with Integers

Concept
PEMDAS Still Rules

Follow the usual order: Parentheses, Exponents, Multiply/Divide (left to right), Add/Subtract (left to right). The only new thing is keeping careful track of signs at every step.

Negative numbers change nothing about the order; they only add sign decisions at each step. The one genuinely new trap is the exponent: an exponent applies to what it touches, so (−2)2(-2)^2 squares −2-2 and gives 44, while −22-2^2 squares only 22 and then takes the opposite, −4-4.

Example
A Multi-Step Problem

Evaluate −3+2×(−4)-3 + 2 \times (-4).

−3+2×(−4)=−3+(−8)multiply first=−11then add\begin{aligned} -3 + 2 \times (-4) &= -3 + (-8) &&\text{multiply first}\\ &= -11 &&\text{then add} \end{aligned}
Side note
Where it goes wrong

Evaluating −32+10-3^2 + 10 as 9+10=199 + 10 = 19. The exponent binds before the negative sign: −32=−9-3^2 = -9, so the value is −9+10=1-9 + 10 = 1. Write (−3)2(-3)^2 when the negative number itself is meant.

Example
With Parentheses and Exponents

Evaluate (−2)2−5(3−7)(-2)^2 - 5(3 - 7).

(−2)2−5(3−7)=4−5(−4)exponent and parentheses=4+20multiply, then add the opposite=24\begin{aligned} (-2)^2 - 5(3 - 7) &= 4 - 5(-4) &&\text{exponent and parentheses}\\ &= 4 + 20 &&\text{multiply, then add the opposite}\\ &= 24 \end{aligned}

Note (−2)2=(−2)(−2)=4(-2)^2 = (-2)(-2) = 4, but −22=−(2×2)=−4-2^2 = -(2\times 2) = -4. Parentheses matter!

Tip

Tip: A negative in parentheses raised to a power, like (−2)2(-2)^2, is different from −22-2^2. Watch where the parentheses are.

Try it
Try it: step by step

Evaluate −2(5−9)2+12÷(−4)-2(5 - 9)^2 + 12 \div (-4).

Show the answer

Parentheses: 5−9=−45 - 9 = -4. Exponent: (−4)2=16(-4)^2 = 16. Multiply and divide left to right: −2×16=−32-2 \times 16 = -32 and 12÷(−4)=−312 \div (-4) = -3. Add: −32+(−3)=−35-32 + (-3) = -35.

Word Problems with Integers

Concept
Translating Situations

Real situations often use negatives: temperatures below zero, money owed (debt), and elevation below sea level. Rising, gaining, and depositing are positive; falling, losing, and spending are negative.

Signed numbers let one rule handle every direction: choose which way is positive, and then every rise, gain or deposit is added and every fall, loss or withdrawal is added as a negative. The payoff is that "how much warmer" and "how much colder" become the same subtraction, the change from one value to another: new−old\text{new} - \text{old}.

Example
Temperature Change
−7−6−5−4−3−2−101234rose 9°dawnnoon
From −6∘-6^\circ up 9∘9^\circ to 3∘3^\circ.

At dawn it was −6∘-6^\circC. By noon the temperature rose 99 degrees. What was the noon temperature? Rising means adding: −6+9=3-6 + 9 = 3. The noon temperature was 3∘3^\circC.

Side note
Where it goes wrong

Computing the change from −6∘-6^\circ to 3∘3^\circ as 3−6=−33 - 6 = -3. The change is new minus old with the sign kept: 3−(−6)=93 - (-6) = 9, a rise of 99 degrees.

Example
Money Owed and Elevation

Money: Maya owes her brother $1212 (that is −12-12) and pays back $55. Her balance is −12+5=−7-12 + 5 = -7, so she still owes $77. Elevation: A diver is 1818 m below sea level (−18-18) and descends 77 more meters. New depth: −18+(−7)=−25-18 + (-7) = -25, so 2525 m below sea level.

Tip

Tip: Choose a positive direction (up, gain, deposit) and make the opposite negative. Then just add the signed amounts.

Try it
Try it: elevation

A hiker stands at 150150 m above sea level; a diver is at −35-35 m. How far apart vertically are they? A submarine at −210-210 m rises 8585 m. Where is it now?

Show the answer

Distance apart is new minus old: 150−(−35)=185150 - (-35) = 185 m. The submarine: −210+85=−125-210 + 85 = -125 m, so 125125 m below sea level.

Going Deeper: Advanced Integer Ideas

Concept
Why (−)(−)=(+)(-)(-) = (+): A Real Proof

The rule “a negative times a negative is positive” is not arbitrary—it is forced by the distributive property, a(b+c)=ab+aca(b+c) = ab + ac. Watch:

(−1)(1+(−1))=(−1)(0)=0.(-1)\big(1 + (-1)\big) = (-1)(0) = 0.

But distributing the same expression gives

(−1)(1)+(−1)(−1)=−1+(−1)(−1).(-1)(1) + (-1)(-1) = -1 + (-1)(-1).

Both equal 00, so −1+(−1)(−1)=0-1 + (-1)(-1) = 0, which means (−1)(−1)(-1)(-1) must be +1+1. Any other value would break arithmetic. Two negatives must make a positive.

Concept
Closure and Properties of the Integers

A set is closed under an operation if combining any two members always lands you back inside the set. The integers are closed under addition, subtraction, and multiplication: adding, subtracting, or multiplying two integers always yields another integer. They are not closed under division, since 3÷2=1.53 \div 2 = 1.5 is not an integer. Other key properties for all integers a,b,ca,b,c:

  • Commutative (add, multiply): a+b=b+aa+b = b+a and ab=baab = ba.
  • Associative (add, multiply): (a+b)+c=a+(b+c)(a+b)+c = a+(b+c).
  • Distributive: a(b+c)=ab+aca(b+c) = ab + ac.
  • Identities: a+0=aa + 0 = a and a×1=aa \times 1 = a.
  • Additive inverse: every integer aa has an opposite −a-a with a+(−a)=0a + (-a) = 0.

Subtraction is not commutative or associative: 5−3≠3−55 - 3 \neq 3 - 5.

Concept
Modular Arithmetic with Negatives

When we divide, the remainder is what is left over. Mathematicians agree that the remainder should always be non-negative, even when the number is negative. So we find a mod na \bmod n by adding or subtracting copies of nn until we land in the range 00 to n−1n-1.

For example, −7 mod 5-7 \bmod 5: add 55 twice, −7+5+5=3-7 + 5 + 5 = 3, so −7≡3(mod5)-7 \equiv 3 \pmod 5. Check: −7=(−2)(5)+3-7 = (-2)(5) + 3, and the remainder 33 sits in {0,1,2,3,4}\{0,1,2,3,4\}. This is exactly how a clock works—“33 hours before 22 o'clock” wraps around to 1111.

Example
Worked Example: Remainders of Negatives

Find −23 mod 7-23 \bmod 7.

−23=q⋅7+r,0≤r<7want non-negative remainder−23+7+7+7+7=5add 7 four times−23=(−4)(7)+5so q=−4, r=5\begin{aligned} -23 &= q \cdot 7 + r, \quad 0 \le r < 7 &&\text{want non-negative remainder}\\ -23 + 7 + 7 + 7 + 7 &= 5 &&\text{add } 7 \text{ four times}\\ -23 &= (-4)(7) + 5 &&\text{so } q = -4,\ r = 5 \end{aligned}

Therefore −23≡5(mod7)-23 \equiv 5 \pmod 7. Notice q=−4q = -4, not −3-3: we round the quotient down so the remainder stays non-negative.

Tip

Tip: For a negative number mod nn, keep adding nn until you reach a value between 00 and n−1n-1. That value is the true remainder.

Concept
Absolute-Value Equations and Inequalities (Preview)

Because ∣x∣|x| measures distance from 00, an equation like ∣x∣=5|x| = 5 has two answers: x=5x = 5 or x=−5x = -5 (both are 55 units away). More generally:

∣x∣=k ⇒ x=k  or  x=−k(k≥0).|x| = k \ \Rightarrow\ x = k \ \text{ or } \ x = -k \quad (k \ge 0).

Inequalities split into two shapes:

  • ∣x∣<k|x| < k means xx is within kk of zero: −k<x<k-k < x < k (a single band).
  • ∣x∣>k|x| > k means xx is farther than kk from zero: x<−kx < -k or x>kx > k (two pieces).

And ∣x∣=−3|x| = -3 has no solution: distance can never be negative.

Example
Worked Example: Solving ∣x−2∣=6|x - 2| = 6
−5−4−3−2−1012345678966
The two numbers at distance 66 from 22.

The bars say the distance from xx to 22 is 66. So x−2x - 2 is either 66 or −6-6:

x−2=6⇒x=8x−2=−6⇒x=−4\begin{aligned} x - 2 &= 6 &&\Rightarrow\quad x = 8\\ x - 2 &= -6 &&\Rightarrow\quad x = -4 \end{aligned}

The two solutions are x=8x = 8 and x=−4x = -4. Check: ∣8−2∣=∣6∣=6|8 - 2| = |6| = 6 and ∣−4−2∣=∣−6∣=6|{-4} - 2| = |{-6}| = 6. Both work.

Concept
Sums of Consecutive Integers

The sum of the integers from 11 to nn has a famous closed form (Gauss's trick: pair the first with the last):

1+2+3+⋯+n=n(n+1)2.1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}.

For example, 1+2+⋯+100=100⋅1012=50501 + 2 + \cdots + 100 = \dfrac{100 \cdot 101}{2} = 5050. This also reveals a neat fact: the sum of any run of consecutive integers that is symmetric about 00, such as −4+(−3)+⋯+3+4-4 + (-3) + \cdots + 3 + 4, is 00, because every term cancels its opposite.

Example
Telescoping Sums

A telescoping sum collapses because inner terms cancel in pairs. Consider

(1−2)+(2−3)+(3−4)+(4−5).(1 - 2) + (2 - 3) + (3 - 4) + (4 - 5).

Rewrite without the grouping and cancel adjacent opposites:

1 − 2+2⏟0 − 3+3⏟0 − 4+4⏟0 − 5=1−5=−4.1 \ \underbrace{-\,2 + 2}_{0}\ \underbrace{-\,3 + 3}_{0}\ \underbrace{-\,4 + 4}_{0}\ -\,5 = 1 - 5 = -4.

Only the very first and very last numbers survive. Telescoping turns a long chain of signed integers into a quick subtraction.

Tip

Big picture: the integer sign rules, closure, and Gauss's formula are the foundation for algebra. Master signed arithmetic now and equations, sequences, and modular patterns all become far easier later.

Why It Works

The sign rules are forced by the distributive property. Since (−2)×0=0(-2)\times0 = 0 and 0=3+(−3)0 = 3 + (-3), we get (−2)(3)+(−2)(−3)=0(-2)(3) + (-2)(-3) = 0, i.e. −6+(−2)(−3)=0-6 + (-2)(-3) = 0, so (−2)(−3)=6(-2)(-3) = 6. Any other value would break the law a(b+c)=ab+aca(b + c) = ab + ac that all of arithmetic relies on. Subtraction as "add the opposite" follows from a−ba - b being the number that added to bb gives aa.

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
Signed number rules
a−b=a+(−b),(−)(−)=+,(−)(+)=−a-b=a+(-b),\qquad (-)(-)=+,\qquad (-)(+)=-

What it means. Subtracting is adding the opposite; two negatives multiply to a positive.

Example. −5−(−3)=−5+3=−2-5-(-3)=-5+3=-2, and (−2)(−4)=8(-2)(-4)=8.

Why it works. (−1)(−1)(-1)(-1) must be +1+1: since (−1)(1+(−1))=(−1)(0)=0(-1)\big(1+(-1)\big)=(-1)(0)=0, and the first piece is −1-1, the second piece must be +1+1 to cancel it.

Tip. Count the negative factors — an even count is positive, an odd count negative.