Count the people in your house. Now count the people in an empty room. Now count how many dollars you have if you owe your friend five. Each answer needs a bigger kind of number than the one before. That is the whole of today.
7 is a counting number. Counting numbers are natural, and every natural number is also whole, integer, rational and real.
Notice: Each number lands in its smallest box, and every bigger box around it lights up too. 7 lives in Natural, but it is also Whole, also Integer. โ4 skips the two inner boxes.
The boxes sit inside each other. A number belongs to its own box and to every box around it.
Three boxes, one inside the other
Start with the numbers you have used since you were three: 1, 2, 3, 4 and on forever. These are the natural numbers, the counting numbers. You can count sheep, steps or Robux with them. You cannot count nothing, and you cannot count a debt.
Now add one number: 0. That gives the whole numbers, 0, 1, 2, 3, โฆ Zero is not a counting number, but it is a perfectly good answer to 'how many?' when the answer is none. Notice the whole numbers contain every natural number. The natural box sits inside the whole box.
Now add the negatives: โ1, โ2, โ3, โฆ Temperatures below zero, floors below ground, money you owe. Whole numbers plus negatives make the integers: โฆ โ3, โ2, โ1, 0, 1, 2, 3 โฆ The integer box holds the whole box, which holds the natural box. Three boxes, one inside the other.
So when someone asks 'is 5 an integer?', the answer is yes, and it is also whole, and also natural. A number belongs to its smallest box and to every box around it. The question we will keep asking all chapter is: what is the smallest box this number fits in? That is the number's home.
- โ
- the natural numbers, 1, 2, 3, โฆ
- W
- the whole numbers, 0, 1, 2, 3, โฆ
- โค
- the integers, โฆ, โ2, โ1, 0, 1, 2, โฆ
- โ
- 'sits inside': every member of the left set is also in the right set
Every natural number is a whole number, and every whole number is an integer. The boxes nest like bowls in a cupboard.
Introduction to rational and irrational numbers
- The first minute, where he lists the integers and shows they include the negatives and zero.
- He then goes one box further out, to rational numbers. That is our next bite: just notice the box exists.
- Every time he says 'integer', ask yourself: which of our three boxes is that?
7 is a counting number. Counting numbers are natural, and every natural number is also whole, integer, rational and real.