Negative Numbers and Linear Equations

Below Zero #

The number line carries on past zero.

The number line carries on below zero, and the minus sign names that side

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Zero is not the end of the scale. It carries on below — that is where negatives live.

3 below zero is −3. The minus sign says which side of zero you are on.

Start at −3 and add 4. Adding warms you up, right past zero, and lands you on 1.

Ordering Negative Numbers #

Bigger means higher, not bigger-looking.

Negatives order by depth below zero, so the bigger-looking number can be the smaller

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−2 sits two rungs below zero. Where does −7 go? Five rungs deeper still.

−7 hangs below −2, so −7 is smaller. Bigger means higher up, not bigger-looking.

Sideways it still holds: −6, then −1, then 3 — position is the whole story.

So −2 > −3: the higher of two numbers is the larger, however they look.

Adding and Subtracting Across Zero #

Walks that pass through zero.

Adding climbs and subtracting drops, and either walk can carry you across zero

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Take away more than you have: 5 − 9 drops nine rungs, straight past zero to −4.

Two falls stack: −3 + (−4) starts below zero and drops 4 more, landing on −7.

Adding still climbs: −3 + 5 rises through zero and comes out at 2.

Subtracting Negative Numbers #

Taking away a debt leaves you better off.

Subtracting a negative adds, because taking away a debt leaves you better off

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Subtract less each row and the answer climbs by 1: 5 − 2 = 3, 5 − 1 = 4, 5 − 0 = 5.

Carry the pattern one row below zero. The answer has to climb by 1 once more.

So 5 − (−1) = 6 and 5 − (−2) = 7. Subtracting a negative adds it on.

Owe 2 and have the debt wiped: you end 2 better off. That is 5 − (−2) = 5 + 2.

Absolute Value #

How far a number sits from zero.

The absolute value of a number is its distance from zero, so it is never negative

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−5 and 5 sit the same distance from zero: five steps, one going each way.

Write that distance as |x|. The bars keep the size, so |x| is never below zero.

Two numbers now. The gap from −2 up to 6 is eight steps of the same line.

The distance between two numbers is |p − q| — subtract either way round.

Multiplying and Dividing Negative Numbers #

Two minus signs multiply to a plus.

Two negatives multiplied give a positive, and dividing follows the same sign rule

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Down each row the left number drops by 1, and the answer climbs by 2.

Carry the pattern below zero. The answer has to climb by 2 once more.

So −1 × −2 is 2, and −2 × −2 is 4. Two negatives multiplied give a positive.

Division reads a multiplication backwards, so the same sign rule settles it.

Algebraic Notation #

A letter holds a number you do not know yet.

A letter is a placeholder for a number you do not know yet

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The scales are perfectly level, so x has to be 5.

Here x + 2 balances 7. The letter still stands for one particular number.

Solving Linear Equations #

Do the same to both sides.

Whatever you do to one side you must do to the other

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You want x all on its own. That 2 is in the way.

Take 2 off both pans and the scales stay perfectly level.

x is alone and the pans are still level, so x = 5.

A two-step one: 3x + 2 = 14. Clear the added 2 first, and both pans lose 2.

That leaves 3x = 12. Three lots of x weigh 12, so one x weighs 4.