### Probability Rules

I say yes in the sense that multiplication and addition are deeper concepts themselves than the Originally Answered: Is there a deeper relationship between multiplication and addition Are the rules for multiplication and addition of significant figures mathematically sound? Multiplication is defined in terms of addition. The concepts of addition, subtraction, multiplication and division are complex abstract taken to be evidence of understanding and high mathematical ability. Equally Use the relationship between multiplication and division as inverses. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that and use the rules for adding, subtracting, multiplying and dividing them. 2. Order of a big difference to the answer. The square.

### Multiplication concept, plus Multiplication and Addition

The commutative and associative laws also hold for multiplication see Box 3—1. The commutativity of multiplication by 2 is also reflected in the equivalence of the two definitions of even number typically offered by children. In addition to these two laws for each operation, there is a rule, known as the distributive law, connecting the two operations.

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A good way to visualize the distributive law is in terms of the area interpretation of multiplication. Then it says that if I have two rectangles of the same height, the sum of their areas is equal to the area of the rectangle gotten by joining the two rectangles into a single one of the same height but with a base equal to the sum of the bases of the two rectangles: The basic properties of addition and multiplication of whole numbers are summarized in Box 3—1.

Page 77 Share Cite Suggested Citation: The order of the two numbers does not affect their sum: When adding three or more numbers, it does not matter whether the first pair or the last pair is added first: The order of the two numbers does not affect their product: When multiplying three or more numbers, it does not matter whether the first pair or the last pair is multiplied first: Page 78 Share Cite Suggested Citation: When multiplying a sum of two numbers by a third number, it does not matter whether you find the sum first and then multiply or you first multiply each number to be added and then add the two products: Subtraction and Division So far we have talked only about addition and multiplication.

It is traditional, however, to list four basic operations: As implied by the usual juxtapositions, subtraction is related to addition, and division is related to multiplication.

The relation is in some sense an inverse one. By this, we mean that subtraction undoes addition, and division undoes multiplication. This statement needs more explanation. Just as people sometimes want to join sets, they sometimes want to break them apart.

If Eileen has eight apples and eats three, how many does she have left? The answer can be pictured by thinking of eight apples as composed of two groups, a group of five apples and a group of three apples.

When the three are taken away, the five are left. Thus subtracting three undoes the implicit addition of three and leaves you with the original amount. More formally, subtracting 3 is the inverse of adding 3.

A second interpretation of addition comes from extending an initial length by a given length: When an original length is extended by a given amount, the final length is the sum of the original length and the length of the extension. The unary view is also useful when discussing subtractionbecause each unary addition operation has an inverse unary subtraction operation, and vice versa.

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The fact that addition is commutative is known as the "commutative law of addition". Some other binary operations are commutative, such as multiplication, but many others are not, such as subtraction and division.

That addition is associative tells us that the choice of definition is irrelevant. When addition is used together with other operations, the order of operations becomes important. In the standard order of operations, addition is a lower priority than exponentiationnth rootsmultiplication and division, but is given equal priority to subtraction.

Units[ edit ] To numerically add physical quantities with unitsthey must be expressed with common units. Innate ability[ edit ] Studies on mathematical development starting around the s have exploited the phenomenon of habituation: This finding has since been affirmed by a variety of laboratories using different methodologies.

Here is an image above which illustrates Todd's idea of a mathematical weave between two axes. The image is titled "Distance" and it uses the distace equation: Also, addition seems to be one-dimensional, while multiplication seems to create two dimensions.

Addition happens along the number line, while multiplication can be graphed along the x and y axis. They say you can't add apples and oranges. In addition you have to find a common denominator before you can add.

This implies the number line again. As soon as two things are on the same dimension they can be added. For example, de Chirico and Baltimore are both physical things and so they can both be photographed together and said to be "added together" in the picture.

But with multiplication there is less restriction.

You don't need a common denominator to multiply two things. The combination creates something new that is not merely more quantity of a common denominator. In pure mathematics 3 x 4 creates a rectangle of area Before there were only lines one dimensionafter multiplication there is area two dimensions.

New space is created. In the example of de Chirico, Baltimore x de Chirico created a new vision of Baltimore colored by de Chirico's own inspiration. No one had seen Baltimore in quite the same way. It is as if a new dimension was opened when these two were combined. Well, I didn't plan to write this much, but it's fun to think about.

Thanks, Todd I also want to thank Todd Smith for his wonderful comments as well.