What Is a Number?

A useful model for real numbers

One very useful way to build intuition about real numbers is to think about distance and direction on a number line. This becomes especially helpful once we start working with signed numbers and addition.

Mathematics contains many kinds of numbers and several rigorous ways to construct them, so “distance from an origin” is better understood as a model than as a universal definition of number. It is a way of seeing numbers that makes certain structure very clear.

Number and numeral are different things

The mark 3 is not the number three in the same way the word “cat” is not an actual cat.

The written mark is a numeral: a representation. The number is the abstract mathematical object that representation refers to.

This is another form-and-value distinction. The numeral is a form. The number is the mathematical object we are trying to represent.

A geometric model: origin, unit, direction

Pick a point and call it the origin. Give that point the coordinate 0. Then choose a unit length and a positive direction.

Once those choices are made, every real number corresponds to a position on the line. Positive and negative signs encode which side of the origin we are on, and absolute value tells us the distance from zero.

The numbers -3 and 3 are different values, but they are the same distance from zero.

Zero and the origin are related, but different

The number zero has a very specific structural role in ordinary arithmetic: it is the additive identity.

What we can choose is which physical or geometric location gets coordinate zero.

Sea level can be zero elevation. The freezing point of water can be zero on one temperature scale. A timeline can declare some event to be time zero. A coordinate system can place its origin wherever it is useful.

Moving the origin changes the coordinates we assign to points. It does not change the underlying distances between those points.

The unit matters too

Coordinates only mean something relative to a scale. A distance of 3 could mean three meters, three feet, three seconds from a reference event, or three abstract units in a mathematical model.

If we change units, the numerical representation may change even when the underlying measured quantity does not.

The measurement has a new numerical form because we changed the unit. The physical length did not suddenly become one hundred times longer.

Why I like the movement model

Once numbers are sitting on a line, addition becomes movement in a really natural way.

That picture does not replace every interpretation of addition. If I am counting three apples and then getting two more apples, “combine two quantities” is probably the more natural story.

But the movement model becomes especially useful when negatives enter the picture. It gives the sign a job: magnitude tells us how far, and sign tells us which direction along the chosen axis.

Different number systems solve different problems

One reason “what is a number?” gets complicated is that the collection of things we are willing to call numbers has grown as mathematics has grown.

• Natural numbers are enough for basic counting.
• Integers let subtraction move freely through zero.
• Rational numbers represent ratios of integers.
• Real numbers give us the continuous real number line.
• Complex numbers extend the reals with another independent direction and make equations such as x² + 1 = 0 solvable.

Each extension preserves a lot of the structure we already know while giving us new things we are allowed to do.

For the advanced reader: from the line to the plane

The movement picture becomes even more interesting when we stop insisting on one dimension.

A real number can be pictured as a coordinate on a line. A complex number a + bi can be pictured as the point (a, b) in a plane.

In that form, complex numbers stop looking like a weird exception bolted onto arithmetic and start looking geometric. Multiplication by i, for example, acts like a quarter-turn in the complex plane.

This is exactly the sort of connection I care about in these notes. A new form does not merely make the old idea prettier. Sometimes it makes a structure visible that was extremely hard to see before.

What I want you to carry forward

• A number and the numeral used to write it are different things.
• Thinking of real numbers as directed positions on a line is a model, not the only definition of number.
• In a coordinate model, we choose an origin, a unit, and a positive direction.
• Zero itself has a fixed algebraic role as the additive identity.
• Changing origin or units can change numerical form while preserving important underlying relationships.
• This movement model gives us a useful foundation for thinking about addition next.