# Understanding Integers

This article explains what integers are. The commutative, associative, and distributive properties of integers are explained. It includes the difference between fractions and
integers.

## Understanding Integers

Integers are real numbers that can be expressed without a fraction or decimal.
Real numbers are part of the complex numbers, which include the imaginary numbers.
Imaginary numbers are numbers that are the square root of a negative number. The natural numbers are the counting numbers (The integers 0, 1, 2, 3, 4, 5, ). Examples of integers are 3, 5, 7, 999, 1009, 202.

An integer is greater than another integer if it has a higher value. In other words, an integer to the right of another integer is greater than that integer. Examples are 5>4, 10>6, and 23>3. If an integer is greater than another integer, then that integer is less than it. Negative integers are integers that have a minus sign like -4, -100, -9765, -83, -292993. Negative integers are greater if the positive integer is less than the other integer. Examples are -4>-5, -10>-90, and -100>-200. When you subtract integers, you are using negative integers. If you subtract a negative integer from a positive integer that is less than it, then you subtract the lesser integer from the larger integer, giving it a minus sign. For example, subtracting 10 from 3 is the same thing as subtracting 3 from 10, equaling 7, then adding a negative sign: -7. It can also be written
-10+3=-7.

Integers must be distinguished from fractions like 3/4, 7/5, 45/23, etc. Also there are mixed numbers that are not integers like 1 3/4, 5 7/8, 100 2/3, etc. Any number with a decimal is also not an integer. A decimal is the same thing as a fraction, it uses tenths, hundredths, thousandths, etc. Examples of decimals are .1 (1/10), .345 (345/1000), and .333 (333/1000). The period at the start of the decimal indicates the place. The first integer to the left of the period is the tenths, the second is the hundredths, the third is the thousandths, etc. Mixed numbers are also decimals. Examples are 1 3/4 =1.75, 100 2/3=100.66666, 5 4/10=5.4, etc. Another name for the integers is whole numbers. The even integers are those divisible by 2 without a remainder: 0, 2, 4, 6, 8, 10, ad infinitum. The odd integers are those not divisible by 2 without a remainder: 1, 3, 5, 7, 9, 11, ad infinitum.

Prime integers are those which have only two divisors without remainders-namely 1 and themselves. Examples are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, ad infinitum. They were proven infinite by Euclid many years ago. Prime numbers are a very important part of the branch of mathematics number theory.

There are five important properties of integers. The properties use the symbols a, b, c, d for any integer, positive or negative. The first is called the commutative property of addition: a+b=b+a. Examples are 7+3=3+7=10 and -7+10=10-7=3. The second property is the commutative property of multiplication: axb=bxa. Examples are 5x3=3x5=15 and -5x3=3x-5=-15. The associative property of addition is the third property: (a+b)+c=a+(b+c). Examples are (3+5)+8=3+(5+8)=8+8=3+13=16 and (-3+5)+7=-3+(5+7)=2+7=-3+12=9. The associative property of multiplication is the fourth property: (axb)xc=ax(bxc). Examples are (3x8)x3=3x(8x3)=24x3=3x24=72 and (-3x9)x4=-3x(9x4)=-27x4=-3x36= -108. The fifth property is the distributive property of integers: ax(b+c)=axb+axc. Examples are 5x(3+9)=5x3+5x9=5x12=15+45=60 and -5x(2+8)=-5x2+-5x8=-5x10=-10+( -40)=-50.

The abacus was the first calculator that added and subtracted integers. Calculators have lessened the burden of addition and subtraction.

Whole Numbers

Number Concepts

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I have a Bachelor of Science in Mathematics and a Classical Studies Minor. I took Spanish in high school
and several languages in college including Latin, ancient Greek, French, and German.