The new number aims to take the pressure off the 111 phone number and to help provide better services for the public.
Police receive more than 2 million calls a year, according to Police Minister Stuart Nash.
Almost 900,000 calls are made to the 111 service. There are around 1.2 million non-emergency calls to other police numbers.
"Just 20 per cent of 111 calls result in an emergency response, known as priority one," Nash said.
He said another 20 per cent are priority two.
"This means at least 60 per cent of calls to the 111 number could be better dealt with on other channels."
Nash said some calls involve complaints about parking, noise - or even cheeky children.
"Calls to 111 should be limited to cases where an emergency is happening now or just happened and there's a threat to life or property. If it's already happened and there's no immediate danger, call 105."
Nash said he hoped the introduction of the new number would make it easier for callers to get in touch with the right part of the police service for the right reasons.
"There are many other ways people can get help with community safety and crime prevention. The *555 number is for urgent road issues. There are online forms, the anonymous Crimestoppers 0800 line and the 1737 mental health support line.
"The new non-emergency number 'ten-five' now makes it so much easier to get the right help from these extra Police," Nash said.
When to call 105 (ten-five)
Your bike has been stolen from the park
Someone has graffitied your wall
You've lost your wallet or phone
Your handbag was stolen at the library
Someone broke your car window
If you are a retailer and someone has shoplifted
If someone has done a petrol drive off
If you want to add information to your existing Police report
You see someone acting suspiciously
Your car has been stolen
You suspect a scam
You suspect drug dealing
Your shop has been broken into
Your house has been burgled (if you think the burglar may still be in your home, do not go inside - contact Police immediately by calling 111)
When to call 111
There is a fire of any sort
You need an ambulance because someone is seriously hurt
Someone is breaking into your house right now
There is a car accident where people might be hurt, or cars are blocking the road
Someone has been assaulted and the offender is still there
You are afraid for your safety and or for those around you
You need an emergency Police response
When to call *555 (mobile phone only)
For urgent but not life-threatening traffic matters that don't need an emergency police response
Continuous poor driving
Traffic congestion, breakdowns and obstructions on the highway
Source: Police