Is this IP address private?

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Checked in your browser against the IANA special-purpose address registries. What you type is not sent anywhere.

The private ranges

10.0.0.0/8Private (RFC 1918)
172.16.0.0/12Private (RFC 1918) — 172.16.0.0 to 172.31.255.255
192.168.0.0/16Private (RFC 1918) — most home routers
fc00::/7IPv6 unique local, usually starting fd

Private is not the same as special-use

Some addresses are neither private nor public: 127.0.0.1 is loopback, 169.254.x.x and fe80:: are link-local, and 100.64.0.0/10 is the shared space providers use for carrier-grade NAT. None of them are routed across the internet.

Why does my computer show a different address from this page?

Your computer usually has a private address on your own network, like 192.168.1.23. Your router swaps it for one public address on the way out, and that public address is what websites see. See yours.