Addition and subtraction

Addition and subtraction are the fundamentals of the fundamental concepts of math. These two operations make up the first building block of mathematics. These operations are opposites as addition gives and subtraction takes away.

Addition is when several numbers (constants or variables) are put together and form a number This number could be more, less, or same as one or both of the original numbers. Any number of items can be added together. The symbol for addition is the plus, which is written like this "+".

Examples of addition equations

1+1=2    4+9=13   1+3+5+7+9+11+13+15+17+19+21+23+25+27+29+31+33+35=324    888+88+8=984

In addition-based word problems, "sum" is a very common word. The sum of some numbers is just the value gotten from adding them all up. For example, the sum of five and four is nine, so if you see a question that says, "Find the sum of four and five," you just calmly answer "Nine".

Subtraction, on the other hand, is when the second number takes away from the first number. Again, this could be any number of items being added together. The symbol for subtraction, the minus (-), is simply a horizontal line.

Examples of subtraction equations

5-3=2    1-1=0    9999-8888-999-112=100    10000-3031=6969

 Similarly to addition, there is also a word that is used to mean that subtraction is being done. If somebody says the term, "the difference of x and y", they are referring to x-y. A question, "Find the difference between 8 and four," the correct answer would be four.

Gauss's summation

Multiplication knowledge required

Let's say somebody asked you to add all of the integers between 1 and 100, inclusive. The expression laid out would be 1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+21+22+23+24+25+26+27+28+29+30+31+32+33+34+35+36+37+38+39+40+41+42+43+44+45+46+47+48+49+50+51+52+53+54+55+56+57+58+59+60+61+62+63+64+65+66+67+68+69+70+71+72+73+74+75+76+77+78+79+80+81+82+83+84+85+86+87+88+89+90+91+92+93+94+95+96+97+98+99+100, which is a pain to write and an even bigger pain to solve. Imagine taking half an hour or maybe even longer to slowly compute every sum. However, using the commutative property of addition that says we can arrange these values in any way we want, we can start searching for patterns. If you start looking, you can identify that 1+100 is 101, and that is same for 99+2 +98+3, 97+4, and so on. Because there are a hundred numbers in this list, there are 50 pairs, with each pair having the value of 101. Now, we can simplify this to a simple multiplication expression, 50*101, and this is very easy to solve, being 5050. This tool can be used to solve many otherwise tedious addition problems.

Addition and subtraction standard algorithm

An algorithm is a set of instruction used to solve a problem. An example of an algorithm is the one that bots use to recommend social media posts/videos and news stories to you based on what you have watched, read, searched, or shared in the past. There is a specific set of instructions in there to complete the task. This algorithm is much simpler, just being used for addition and subtraction equations. This algorithm involves you lining up the two numbers and adding the digits up starting from the smallest and going to the largest. In the case of an addition equation where for one two digits exceed ten, the ten will be "carried" to the next digit. Because of this, the next digit will be one higher than it usually is. If the sum is twenty more, then the next digit will increase by two, if thirty, three, etc. Similarly, for subtraction, if a digit is smaller than the one it is subtracted by, the next digit result will be taken away in the same way as in addition. If you are new to this, you are probably very confused by this, so below will be figures with explanations. The below figure lines up the digits of the problem 17+15.

  17

+15

_____ 

17 is the first value, and 15 is the second. The long horizontal line below the two numbers indicates that you will write the answer below the line. Using the standard algorithm, we start at the smallest value, which will always be the rightmost one according to place value rules. 7+5 is twelve, so then, a ten will carry over to the next digit. To remember this, we place a little "1" to remind to add 1 to the next digit. We can also write 2 under the ones place because now we are sure of that digit. 

  1

  17

+15

_____

     2

Finally, we can add the three ones together and get 3. Therefore, 17+15 is 32. Next let's try a subtraction problem using the standard algorithm.

 91

-48

_____

Now, we can do the first digit. 8 is a larger number than one, so 1 will have to borrow 10 from 91. Then, it will become 11-8, which is 3.

 81

-40

_____

    3

Now, 8-4 is 4, so the final answer to 91-48 is 43.

Review and challenge

Here is a crash-course of this section. You can use this to review, or you can directly read this if you are in a hurry or think you already know this.

-Addition (+) is two things put together. The put-together value is called the sum of the separate values.

-Subtraction is the second value taking itself away from the first value. The result is called the difference of the separate values.

-Gauss's summation states that for 1+2+3...+n, the value is (n+1)(n/2).

Challenge questions

These are designed to have creativity involved and may still be interesting to people who have mastered addition.

-Can you create a word problem based on a real-world situation that tests somebody's knowledge on Gauss's summation?

-Can you find the value of 75+76+77+78...+100? What about 999+1000+1001+...10000?

-Can you create a problem with both addition and subtraction and with over ten terms? Solve this problem without a calculator and see how fast somebody else can solve it.

-Create an addition or subtraction equation with ten-digit numbers and use the standard algorithm to solve it. Then, make it have more terms or more digits. You can also challenge other people with your problem.