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Numberline

Counting Basics

The numberline is the most important tool in math for counting. It is used to build other counting tools late like the Cartesian plane, unit circle and others. A number line has numbers on it evenly spaced like a ruler. The little lines that point to the number are called “tick marks” or “graduations”. A number line can have as many or as few numbers as you want. Since we can have negative numbers they can be on the line as well. A number line showing numbers from –10 to 10 is shown below:

Number Line from −10 to 10 Negative numbers in red, positive numbers in blue, and zero in black. −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 Negative numbers Positive numbers

This numberline has positive numbers colored in blue and negative numbers colored in red, but 0 is black. Zero is in the middle of all numbers so it is neutral being neither + or –. This concept is proven+ later. If we are adding numbers like \(2 + 3 = 5\) we use plus sign “+”. The numbers being added are called addends or sums, the answer is called the sum, result, or total. The opposite of addition is called subtraction. Addition and subtraction are reciprocal operations of each other. Here is a short proof: \(2 + 2 = 4\) because \(4 – 2 = 2\). Teachers call this “proving your work”. When we work backwards from the answer to the starting number this is called “working backwards”.

We use the reciprocal operation to work backwards. Starting with the answer 4 use the opposite of “ + ” sign and instead use “ – ” sign. We subtract 2 away instead of add 2. This can be shown on a number line:

Number Line from −10 to 10 Negative numbers in red, positive numbers in blue, and zero in black. −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 +2 -2

There is a red dot at 2 and blue dot at 4. The curve shows we added “ +2 ” units. A number can be negative or positive. If negative we use a minus sign like this –2. Negative numbers are also called “signed numbers” because they have a minus sign. This shows adding in blue and subtracting in red. Adding and subtraction are additive inverses of each other.

Numbers never end so we can make a number line shorter or longer. Infinity is not a number but a concept that means the set of numbers never ends. The simplest explanation of infinity is to start at ) and add + 1 + 1 + 1... forever. No matter what the largest number you can think of is you could always make it bigger by adding 1 to it.

If we walk to say a number line never ends we can put the infinity symbol " ∞ " at the end of the positive numbers. If we want to say the numbers never end in the negative direction we can use the negative infinity symbol " -∞ " in front of the negative numbers.

To perform addition: place a dot on the starting number and move the dot by the addend (the number we are adding) to the answer. If we add \(2 + 2 = 4\) we place a dot at 2 then move 2 units to the right stopping at 4. We count until we are at the right place. Sometimes we draw a curve from the first dot to the second dot. There are many ways to use a number line:

Number Line from −6 to 7 Negative numbers in red, positive numbers in blue, with curved arrows showing +2 and -2. -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 +2 -2

We subtract two numbers that are the same like 2–2 = 0 we get 0. This is also the same as adding a two numbers that have the same value but different signs like –2+2 = 0. When the numbers are the same value but have different value signs they cancel. In adding to cancel is to equal 0. When two sums cancel they are called additive inverses. – 4 and 4 are additive inverses. If we add them they cancel to equal 0. If we are multiplying we are adding by groups. We do this to make equations shorter and faster to perform. If we are powering we are adding groups of groups of numbers.

Interval Notation

You can graph an inequality like you can graph an equation using a number line. \( 3+5 > x \) is graphed as shown below. Solve the inequality by combining all constants placing a circle at the correct number. The correct number is 8 because \( 3 + 5 = 8 \). Note than is used a “more than sign” but \( 8 > x \) is really \( x < 8 \) . It is a less than so it points to the left. We state the solution in terms of whether the variable is less or more not the sum.

The solution is all numbers less than “8”. We can write the answer using “interval notation”: \( (a, b) = { x | a < x < b } \) \( 3 + 5 > x = (–∞,8)\) in the form (a, b), this means The number is more than – ∞ and less than 8. \( 3 + 5 ≥ x \) is rewritten as x ≤ 8. It uses the inequality sign “less than or equal to”. Use a dot instead of a circle.

You can have two inequalities at once called compound inequalities. If x is less than 8 and more than 2: 2 < x < 8, or 8 > x > 2, or (2,8). It is graphed as:

It uses two location marks, one at 2 and one at 8, both are circles because x can not be 2 or 8. x is any number in between 2 and 8. If x could be equal to 8 then we would use “≤”... 2 < x ≤ 8 has a dot at 8:

2 < x ≤ 8 is a line segment that shows a single distance. If we have 8 < x < 2 (also 2 > x > 8) then we have a gap:

If x can be 2 then we need to use a compound sign “less than and equal” ≤ or “more than or equal” ≥:

Now it has a dot at 2 instead of a circle. This can be written 2 ≤ x < 8 or 8 > x ≥ 2, normally signs point to the left. Signs point to the smaller number. We usually right the smaller number on the left and bigger number on the right but not always. This would be written as [2,8) because brackets include the number and parentheses exclude it. –∞ and ∞ always use parentheses. You can have [0,∞), which is all positive numbers, or (–∞,0), all negatives, but you could never write it as [–∞,0] or [0,∞] because infinity is not a number so can not be included since you can never reach it.