31.01 - Numbers

Positive numbers (non-negative)

Positive numbers are numbers that are bigger than 0. The following numbers are examples of positive numbers:
$1,2,6,12,43,55,182,1042,45544$
If you’d draw a line with 0 in the middle, positive numbers would be on the right of 0

Negative numbers (non-positive)

Negative numbers are numbers that are smaller than 0. The following numbers are examples of negative numbers (note that they all start with a minus(-) sign:
$-4,-7,-18,-24,-65-73,-273,-584,-3499349.$
If you’d draw a line with 0 in the middle, positive numbers would be on the left of 0

Zero (non-positive and non-negative)

0 (zero) is a special number, it’s neither positive or negative. Zero lays in between positive and negative.

Integers

Integers, or whole numbers, are numbers like 1 2 45 765 2343, -5, -76, -1023 and also 0. Integers are split into 3 categories:

Decimals

A decimal number is a number with a comma(or dot) inside it, they’re not whole numbers. Numbers like: $5.5$, $10.734$, and $0.235$ are decimal numbers.

The numbers behind the comma(or dot) are called decimals; $4.15$ is a decimal number with 2 decimals in it.

The opposite and absolute value

The opposite of a number is its positive/negative counterpart.
To be more mathematical about it: If you add a number and its opposite number, the result will always be $0$
The mathematical notation for the absolute value is $-n$ where n is any number
For example:
The opposite of $19$ is $-19$ The opposite of $-402$ is $402$

The absolute value is the positive counterpart of a number.
So every positive number is its own absolute value, but the opposite isnt.
The mathematical notation for the absolute value is $|n|$ where n is any number
For example: The absolute value of $-20$ $20$, or $|-20|=20$ The absolute value of $55$ is 55 or $|55|=55$

Ranking of numbers

If you’d take the number $4832$, you could say that it consists of:

• $4$ thousands
• $8$ hundreds
• $3$ tens
• 2 singles The same could be done with $0.1258$:
• $0$ singles
• $1$ tenths
• $2$ hundredths
• $5$ thousandths
• $8$ ten thousandths

As you can see positive and negative have both the same name, but negatives have -ths suffixed at the end.

Rounding

Rounding a number means you either round an integer to an integer with a higher rank, a decimal number to an integer, or a decimal number to a decimal number with less decimals.

The reason why you would round a number is so simplify it visually or when extra numbers are redundant.

The way you round a number is by first deciding where you want to round to.

• rounding to a whole number
• rounding to 3 decimals
• rounding to a ten

After that you look to the number on the right of the chosen number and see if its in either $0-4$ or $5-9$.

If its the former, it means the number becomes 0 (rounding down) and nothing else happens. If it’s the latter, the number also becomes 0, but the number on the left goes up by 1.

For example:

• $0.147$ rounded to two decimals becomes $0.15$ because $7$ is between 5 and 9.
• $0.143$ rounded to two decimals becomes $0.14$ because $3$ is between 0 and 4.
• $0.145$ rounded to two decimals becomes $0.15$ because $5$ is between 5 and 9.

When you round a number, you use the approximate sign($\approx$) instead of the equals($=$) sign.

Sometimes you have to decide to either force to round down (floor) or round up (ceiling).

A way to note this is: $\lfloor 5\rfloor$ for floor, and $\lceil 5\rceil$, and both combined is $\lfloor 5\rceil$, which rounds to the nearest integer(normal rounding).

Real numbers

Both rational and irrational numbers are on the number line.

Rational numbers

Rational numbers are integers that can be written like a fraction (a/b) whose denominator is not zero. For example: $5$ can be written as $\frac{5}{1}$ and thus is a rational number.

Decimals in rational numbers are either of two things:

• a finite number of decimals (can also be 0)
• an infinite number of decimals which have a repeating pattern to them

Rational number’s decimals are sometimes written with a bar like this: $5.\overline{437382}$.

Irrational numbers

Irrational numbers are numbers that have an infinite number of decimals which do not repeat. They cannot be written like a fraction. Numbers like $\sqrt{2}$, $\sqrt{3}$, and $\pi$ are irrational. Irrational number’s decimals are usually written with dots at the end like this: $\mathrm{2.1453728...}$

next chapter

The next chapter will go over Ordering of integers